English+GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Gruppenoperation</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Gruppenoperation</h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">In der <a href="Mathematik" title="Mathematik">Mathematik</a> gehört zu einer Gruppenoperation, -aktion oder -wirkung eine <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> als „aktiver“ Teil und eine <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> als „passiver“ Teil. Die Operation, Aktion oder Wirkung eines Elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist eine Transformation (<a href="Bijektive_Funktion" title="Bijektive Funktion">bijektive</a> <a href="Selbstabbildung" title="Selbstabbildung">Selbstabbildung</a>) dieser Menge. Dabei operieren die Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,h\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23eaf33f751fae514c9b3e99dfb98a7718241dda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.156ex; height:2.509ex;" alt="{\displaystyle g,h\in G}" loading="lazy"> auf den Elementen der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in der Weise, dass die Aktion des Produkts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1263fee243ae7e6c4a9ba07670c061b1c31f510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.65ex; height:2.509ex;" alt="{\displaystyle g*h}" loading="lazy"> der <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">Hintereinanderausführung</a> der Einzelaktionen entspricht.

Die operierende Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> wird Transformationsgruppe genannt. Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> zusammen mit der Operation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Menge.

Ist bei der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> zusätzliche Struktur von Bedeutung, sei es <a href="Algebra" title="Algebra">algebraische</a>, <a href="Geometrie" title="Geometrie">geometrische</a> oder <a href="Topologie_(Mathematik)" title="Topologie (Mathematik)">topologische</a>, wird eine Gruppenoperation nur dann als zulässig erachtet, wenn sie diese Struktur bewahrt.

Die Gruppenoperation ermöglicht es in Algebra, Geometrie und vielen anderen Bereichen der Mathematik, die <a href="Symmetrie_(Geometrie)" title="Symmetrie (Geometrie)">Symmetrien</a> von Objekten mit Hilfe von <a href="Symmetriegruppe" title="Symmetriegruppe">Symmetriegruppen</a> zu beschreiben. Hier steht die Untersuchung der Menge, auf der die Operation wirkt, im Vordergrund.

Andererseits kann die Operation einer vorgegebenen Gruppe auf geeignet gewählten Mengen in der <a href="Gruppentheorie" title="Gruppentheorie">Gruppentheorie</a> wichtige Informationen über die Struktur der operierenden Gruppe liefern. In diesem Fall steht die Untersuchung der operierenden Gruppe im Vordergrund.

<div class="mw-heading mw-heading2"><h2 id="Einführendes_Beispiel:_Würfelgruppe_und_Raumdiagonalen">Einführendes Beispiel: Würfelgruppe und Raumdiagonalen</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABCDEFGH}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb07c6eee1fa662b1c38d9d2e5a0e84aff07a989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.604ex; height:2.176ex;" alt="{\displaystyle ABCDEFGH}" loading="lazy"> seien die Ecken eines Würfels in der üblichen Bezeichnung, d. h., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABCD}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/412b7d8df4db6ca8093d971320c405598c49c339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EFGH}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle EFGH}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4eb6bb9952c2c438b42bcfb4e9a79057d3cec779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.407ex; height:2.176ex;" alt="{\displaystyle EFGH}" loading="lazy"> sind gegenüberliegende Flächen (siehe erstes Bild). Die Drehung des Würfels um die Achse, die die Mittelpunkte dieser beiden Flächen verbindet (zweites Bild), induziert die folgende Vertauschung der Ecken:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\;\,\mapsto \;\,B\;\,\mapsto \;\,C\;\,\mapsto \;\,D\;\,\mapsto \;\,A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9caa9137b7503258e6796ceeaa422b3b3c966840.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:31.655ex; height:2.176ex;" alt="{\displaystyle A\;\,\mapsto \;\,B\;\,\mapsto \;\,C\;\,\mapsto \;\,D\;\,\mapsto \;\,A}" loading="lazy">   und gleichzeitig</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\;\,\mapsto \;\,F\;\,\mapsto \;\,G\;\,\mapsto \;\,H\;\,\mapsto \;\,E.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75e3a1288d4644250872168fd693bdb51dc8002e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:32.544ex; height:2.176ex;" alt="{\displaystyle E\;\,\mapsto \;\,F\;\,\mapsto \;\,G\;\,\mapsto \;\,H\;\,\mapsto \;\,E.}" loading="lazy"></dd></dl>

Durch die Drehung werden auch (gleichzeitig) die vier Raumdiagonalen vertauscht, nämlich

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AG\mapsto BH\mapsto CE\mapsto DF\mapsto AG.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afe476b1d8a1dab841124adb9349ced914365cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:33.277ex; height:2.176ex;" alt="{\displaystyle AG\mapsto BH\mapsto CE\mapsto DF\mapsto AG.}" loading="lazy"></dd></dl>

Eine weitere Symmetrieabbildung, die Spiegelung an der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABGH}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABGH}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18b807983ad6a672039d3113878a988a5824d2a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.397ex; height:2.176ex;" alt="{\displaystyle ABGH}" loading="lazy"> (viertes Bild), lässt die zwei Raumdiagonalen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AG}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AG}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d06213a8ade7e2bbe5031079213666b4b9d5e18c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.57ex; height:2.176ex;" alt="{\displaystyle AG}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BH}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BH}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20994bf9422bb445fddaad2a4205cabd88aa1536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.828ex; height:2.176ex;" alt="{\displaystyle BH}" loading="lazy"> fest und vertauscht die anderen zwei:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle CE\mapsto DF}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83f82bbf19f9944d1819e81c3e6da18359bde148.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.821ex; height:2.176ex;" alt="{\displaystyle CE\mapsto DF}" loading="lazy">   und   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF\mapsto CE.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DF\mapsto CE.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59019b1c2c842f4b36d8da43006b92854317abaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.468ex; height:2.176ex;" alt="{\displaystyle DF\mapsto CE.}" loading="lazy"></dd></dl>

Es gibt aber auch Symmetrieabbildungen des Würfels, die die Raumdiagonalen nicht untereinander vertauschen, nämlich die Punktspiegelung am Mittelpunkt (drittes Bild): Sie entspricht

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mapsto G\mapsto A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ba9ab5a7815c49e65f32eaca9959e83d6b5793f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.541ex; height:2.176ex;" alt="{\displaystyle A\mapsto G\mapsto A}" loading="lazy">   und gleichzeitig</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\mapsto H\mapsto B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0176c0cc552ec4a637b03a4a1900f4ee824ccf73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.82ex; height:2.176ex;" alt="{\displaystyle B\mapsto H\mapsto B}" loading="lazy">   und gleichzeitig</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C\mapsto E\mapsto C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/009cef45bf046c640e2ded5c888f7117607be0ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.536ex; height:2.176ex;" alt="{\displaystyle C\mapsto E\mapsto C}" loading="lazy">   und gleichzeitig</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\mapsto F\mapsto D.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f11f668d5d18d8b2f462416b779464cdb9f8fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.464ex; height:2.176ex;" alt="{\displaystyle D\mapsto F\mapsto D.}" loading="lazy"></dd></dl>

Dabei wird jede einzelne Raumdiagonale, wenn auch gespiegelt, so doch auf sich selbst abgebildet.

Man sagt: Die Gruppe der Symmetrieabbildungen des Würfels (genannt die „<a href="W%C3%BCrfelgruppe" class="mw-redirect" title="Würfelgruppe">Würfelgruppe</a>“) operiert auf der Menge der Ecken, auf der Menge der Kanten, auf der Menge der Raumdiagonalen etc. Um diese Gruppe zu erfassen, werde im Folgenden der Fokus auf die Permutationen der Raumdiagonalen gerichtet.

Es gibt zu jedem Paar von Raumdiagonalen eine Ebenenspiegelung (in der Abbildung zum Paar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CE}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle CE}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34ad8d379d96dc8b0c31a7f8006b5768ff0b2fe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.542ex; height:2.176ex;" alt="{\displaystyle CE}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DF}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c00d6204f2d95ede007728da3d08679156a5340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.665ex; height:2.176ex;" alt="{\displaystyle DF}" loading="lazy">), die diese beiden vertauscht und alle anderen Raumdiagonalen fest lässt, nämlich die Spiegelung an derjenigen Ebene, die die festbleibenden Raumdiagonalen enthält. Eine solche paarige Vertauschung heißt <a href="Zyklische_Permutation" title="Zyklische Permutation">Transposition</a>, und diese Transpositionen <a href="Symmetrische_Gruppe#Erzeugende_Mengen" title="Symmetrische Gruppe">erzeugen</a> die ganze symmetrische Gruppe der Permutationen der (vier) Raumdiagonalen. Da es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4!=24}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4!=24}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b31e59bf87c098f00b2c9aad48fe5f85fc86263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.233ex; height:2.176ex;" alt="{\displaystyle 4!=24}" loading="lazy"> dieser Permutationen gibt und genau zwei Symmetrieabbildungen, die alle Raumdiagonalen festlassen (nämlich die Identität und die oben genannte Punktspiegelung), kann man schließen, dass es insgesamt

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 24\cdot 2=48}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8531f49885c08c050675fad78f595aa2305e7a29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.59ex; height:2.176ex;" alt="{\displaystyle 24\cdot 2=48}" loading="lazy"></dd></dl>

Symmetrieabbildungen des Würfels gibt, ohne jede von ihnen einzeln zu kennen. (Für eine genauere Analyse der Gruppenstruktur siehe <a href="Oktaedergruppe" title="Oktaedergruppe">Oktaedergruppe</a>.)

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="(Links-)Aktion">(Links-)Aktion</h3></div>

Eine (Links-)Operation, (Links-)Aktion oder (Links-)Wirkung einer Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> auf einer Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist eine <a href="Zweistellige_Verkn%C3%BCpfung#Äußere_zweistellige_Verknüpfungen_erster_Art" title="Zweistellige Verknüpfung">äußere zweistellige Verknüpfung</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright \colon G\times X\to X;\qquad (g,x)\mapsto g\triangleright x,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/268bfb7959fcb99b748028b713a9f31c7481924e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.307ex; height:2.843ex;" alt="{\displaystyle \triangleright \colon G\times X\to X;\qquad (g,x)\mapsto g\triangleright x,}" loading="lazy"></dd></dl>

mit folgenden Eigenschaften:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\triangleright x=x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c30bfab780d8cabb7eda53845d26459c72d0c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.036ex; height:1.843ex;" alt="{\displaystyle e\triangleright x=x}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> das <a href="Neutrales_Element" title="Neutrales Element">neutrale Element</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ist     („Identität“),</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g*h)\triangleright x=g\triangleright (h\triangleright x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e94b8eeebdaf80edd30e102089be57be1c5a9b92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.065ex; height:2.843ex;" alt="{\displaystyle (g*h)\triangleright x=g\triangleright (h\triangleright x)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h\in G,x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,h\in G,x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03ac5804321be824facca0a8fd6c2e72c2e839da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.341ex; height:2.509ex;" alt="{\displaystyle g,h\in G,x\in X}" loading="lazy">     („Verträglichkeit“).</li></ol>

Man sagt dann, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> operiert (von links) auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, und nennt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> zusammen mit dieser Gruppenoperation eine (linke) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Menge.

Aus den beiden Forderungen folgt, dass für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> die Transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{g\,\triangleright }\colon X\to X,\;x\mapsto g\triangleright x,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta _{g\,\triangleright }\colon X\to X,\;x\mapsto g\triangleright x,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a0853e18bf940dc3487f2cb96454489119853fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.122ex; height:2.843ex;" alt="{\displaystyle \vartheta _{g\,\triangleright }\colon X\to X,\;x\mapsto g\triangleright x,}" loading="lazy"> eine <a href="Bijektivit%C3%A4t" class="mw-redirect" title="Bijektivität">Bijektion</a> (mit der Umkehrabbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{g\,\triangleright }^{-1}=\vartheta _{g^{-1}\triangleright }\colon X\to X,\,x\mapsto g^{-1}\triangleright x}">

</mrow>

</mrow>

</msubsup>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta _{g\,\triangleright }^{-1}=\vartheta _{g^{-1}\triangleright }\colon X\to X,\,x\mapsto g^{-1}\triangleright x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d7dba84c79f82d6204a6aaf43d19522959836dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.842ex; height:3.509ex;" alt="{\displaystyle \vartheta _{g\,\triangleright }^{-1}=\vartheta _{g^{-1}\triangleright }\colon X\to X,\,x\mapsto g^{-1}\triangleright x}" loading="lazy">) auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist.

<ul><li>Im Falle endlicher Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> spricht man von einer <a href="Permutation" title="Permutation">Permutation</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">: Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> liegt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{g\,\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta _{g\,\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc91646804c6d8d3963947a4819dfad912cc5b29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.604ex; height:2.843ex;" alt="{\displaystyle \vartheta _{g\,\triangleright }}" loading="lazy"> in der <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrischen Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy">. Die Aktionen der Gruppenelemente lassen sich also durch <a href="Permutationsdarstellung" class="mw-redirect" title="Permutationsdarstellung">Permutationen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> „darstellen“</a>.</li>

<li>Bei unendlicher Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist der Begriff der Permutation (für eine bijektive Selbstabbildung auf einer Menge) unüblich. Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist im unendlichen Falle in der Regel mit einer (algebraischen, geometrischen oder topologischen) Struktur versehen, und von den Gruppenaktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{g\,\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta _{g\,\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc91646804c6d8d3963947a4819dfad912cc5b29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.604ex; height:2.843ex;" alt="{\displaystyle \vartheta _{g\,\triangleright }}" loading="lazy"> wird die Verträglichkeit mit diesen Strukturen verlangt („Automorphismen“): Die Aktionen der Gruppenelemente lassen sich dann als gewisse <a href="Darstellungstheorie" title="Darstellungstheorie">Automorphismen auf dem Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> „darstellen“</a>, je nachdem, worum es sich bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> handelt, soll heißen, welcher <a href="Kategorientheorie" title="Kategorientheorie">Kategorie</a> das <a href="Kategorientheorie#Kategorie" title="Kategorientheorie">Objekt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> angehört. Auf der Trägermenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> wird ein <a href="H%C3%BCllensystem" class="mw-redirect" title="Hüllensystem">Hüllensystem</a> induziert (bspw. das Hüllensystem (der Verband) der Zwischenkörper), dessen Hüllenoperator (Hülleneigenschaften) durch die Kategorie, welcher das Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> angehört, bestimmt ist und mit dem sich die Gruppenoperation ebenfalls verträgt.</li>

<li>Wenn im Folgenden also von der symmetrischen Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy"> oder von Permutationen die Rede ist, so darf stillschweigend angenommen werden, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> endlich ist. Gruppenoperationen auf endlichen Mengen lassen sich als Teil der <a href="Darstellungstheorie_endlicher_Gruppen" title="Darstellungstheorie endlicher Gruppen">Darstellungstheorie endlicher Gruppen</a> auffassen, vgl. hierzu das Beispiel zur <a href="#Darstellung_durch_Permutationsmatrizen">Darstellung durch Permutationsmatrizen</a>.</li>

<li>Doch alle Begrifflichkeiten und Aussagen gelten <a href="Mutatis_mutandis" class="mw-redirect" title="Mutatis mutandis">mutatis mutandis</a> auch für den Fall einer unendlichen Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> mit einer geeigneten Struktur, wenn anstelle der symmetrischen Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy"> die Automorphismengruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Aut} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Aut} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e84ea9d964acce4ed25be1f7b1f8ef8c70bf04f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.73ex; height:2.843ex;" alt="{\displaystyle \operatorname {Aut} (X)}" loading="lazy"> betrachtet wird und die Gruppenoperation sowie die Gruppenaktionen mit der auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Aut} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Aut} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e84ea9d964acce4ed25be1f7b1f8ef8c70bf04f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.73ex; height:2.843ex;" alt="{\displaystyle \operatorname {Aut} (X)}" loading="lazy"> vorliegenden Struktur verträglich sind. Beispiele hierfür gibt die allgemeine <a href="Darstellungstheorie" title="Darstellungstheorie">Darstellungstheorie</a>, doch auch der <a href="#Beispiele">Abschnitt Beispiele</a> vermittelt hierfür einen ersten Eindruck.</li></ul>

</div>

Des Weiteren folgt erneut aus den Operationseigenschaften, dass eine Gruppenoperation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> mit einem <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a88121f04a342077282bd71e1cfda9c6727c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \Theta _{\triangleright }}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> in die <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrische Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {Sym} (X),\circ )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\operatorname {Sym} (X),\circ )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6e17bb04f3ae554b7abd684a0f022ce667eca33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.251ex; height:2.843ex;" alt="{\displaystyle (\operatorname {Sym} (X),\circ )}" loading="lazy"> gleichgesetzt werden kann:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\Theta _{\triangleright }\colon &(G,*)&\longrightarrow &(\operatorname {Sym} (X),\circ )\\&g&\mapsto &[\vartheta _{g\,\triangleright }:x\mapsto g\triangleright x]\end{matrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7408e8fe8a676ebdc42d5b8d1455a96448cb9110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.286ex; height:6.509ex;" alt="{\displaystyle {\begin{matrix}\Theta _{\triangleright }\colon &(G,*)&\longrightarrow &(\operatorname {Sym} (X),\circ )\\&g&\mapsto &[\vartheta _{g\,\triangleright }:x\mapsto g\triangleright x]\end{matrix}}}" loading="lazy"></dd></dl>

Dieser Gruppenhomomorphismus heißt eine <a href="Permutationsdarstellung" class="mw-redirect" title="Permutationsdarstellung">Permutationsdarstellung</a> der Gruppe, da seine Wirkung auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> durch Permutationen dargestellt wird.

<div class="mw-heading mw-heading3"><h3 id="Rechtsaktion">Rechtsaktion</h3></div>

Analog zur Linksoperation ist eine Rechtsoperation, -aktion oder -wirkung eine äußere zweistellige Verknüpfung

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleleft \colon X\times G\to X;\qquad (x,g)\mapsto x\triangleleft g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf36fb526e75b4f3f71d2f1019fff81d88835fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.66ex; height:2.843ex;" alt="{\displaystyle \triangleleft \colon X\times G\to X;\qquad (x,g)\mapsto x\triangleleft g}" loading="lazy"></dd></dl>

mit

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\triangleleft e=x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/568846da9b831bf63ed51ebba30bead9c0d3ec26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.036ex; height:1.843ex;" alt="{\displaystyle x\triangleleft e=x}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> und das neutrale Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\triangleleft (g*h)=(x\triangleleft g)\triangleleft h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa3e626be92cd01c291946d8145b93424b7b237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.065ex; height:2.843ex;" alt="{\displaystyle x\triangleleft (g*h)=(x\triangleleft g)\triangleleft h}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h\in G,x\in X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,h\in G,x\in X.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d3654d6b4563ad01fdc6620786374b6ed3f045c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.987ex; height:2.509ex;" alt="{\displaystyle g,h\in G,x\in X.}" loading="lazy"></li></ol>

Der Unterschied zwischen Links- und Rechtsoperationen liegt in der Art und Weise, wie Verknüpfungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1263fee243ae7e6c4a9ba07670c061b1c31f510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.65ex; height:2.509ex;" alt="{\displaystyle g*h}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> operieren. Bei einer Linksoperation operiert zuerst <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> und dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">, während bei einer Rechtsoperation die Reihenfolge umgekehrt ist.

Aus einer Rechtsoperation lässt sich eine Linksoperation konstruieren, indem man die Operation als Linksoperation der <a href="Gruppe_(Mathematik)#Gegengruppe" title="Gruppe (Mathematik)">Gegengruppe</a> schreibt, oder auch, indem statt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> von links <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ae77eeb0200a3b0e26ff4b251fb845ca1b385c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:3.009ex;" alt="{\displaystyle g^{-1}}" loading="lazy"> von rechts operiert. Zu jeder Rechtsoperation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleleft }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleleft }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f51c6780e67750600191c48af0a9a2af862c5e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleleft }" loading="lazy"> gibt es eine Linksoperation

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright \colon G\times X\to X;\qquad (g,x)\mapsto g\triangleright x:=x\triangleleft g^{-1},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e15feeb8dfaffe4ed795e1d4bb9931e99c203640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.028ex; height:3.176ex;" alt="{\displaystyle \triangleright \colon G\times X\to X;\qquad (g,x)\mapsto g\triangleright x:=x\triangleleft g^{-1},}" loading="lazy"></dd></dl>

denn

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\triangleright x=x\triangleleft e^{-1}=x\triangleleft e=x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab968d76ddc146a101af8e3fcedb1b5336e89041.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.782ex; height:2.676ex;" alt="{\displaystyle e\triangleright x=x\triangleleft e^{-1}=x\triangleleft e=x}" loading="lazy"></dd></dl>

und

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(g*h)\triangleright x&=x\triangleleft (g*h)^{-1}=x\triangleleft (h^{-1}*g^{-1})\\&=(x\triangleleft h^{-1})\triangleleft g^{-1}=(h\triangleright x)\triangleleft g^{-1}=g\triangleright (h\triangleright x).\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd57d333ce2ecc99cb6e44c1a4ff759d53fd4446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.63ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}(g*h)\triangleright x&=x\triangleleft (g*h)^{-1}=x\triangleleft (h^{-1}*g^{-1})\\&=(x\triangleleft h^{-1})\triangleleft g^{-1}=(h\triangleright x)\triangleleft g^{-1}=g\triangleright (h\triangleright x).\end{aligned}}}" loading="lazy"></dd></dl>

Analog lässt sich eine Links- in eine Rechtsoperation umwandeln. Da sich Links- und Rechtsoperation im Wesentlichen nicht unterscheiden, werden ab hier nur noch Linksoperationen betrachtet.

Bei der Betrachtung der Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a88121f04a342077282bd71e1cfda9c6727c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \Theta _{\triangleright }}" loading="lazy"> ist allerdings Vorsicht geboten: Bei einer Linksoperation ist sie ein Gruppenhomomorphismus, weil Funktionssymbole üblicherweise auch linksseitig notiert werden: „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">“. Bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f61ca7838709fbae07dce9c0d513770f10cfae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" loading="lazy"> operiert daher zuerst <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> und anschließend <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Das entspricht der Reihenfolge bei einer Linksoperation gemäß der Eigenschaft der Verträglichkeit. Operiert die Gruppe aber von rechts, so handelt es sich um einen Antihomomorphismus von Gruppen, es sei denn, man trifft Vorkehrungen, wie oben geschildert, um dies zu verhindern.

<div class="mw-heading mw-heading2"><h2 id="Weitere_Begriffe">Weitere Begriffe</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Homomorphismen_zwischen_G-Mengen">Homomorphismen zwischen G-Mengen</h3></div>

Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> eine weitere Menge mit einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Linksoperation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \star }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd316a21eeb5079a850f223b1d096a06bfa788c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }" loading="lazy"> ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon X\to Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56ddf345aff7df103700ce620d30554abe6d78f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.921ex; height:2.676ex;" alt="{\displaystyle \varphi \colon X\to Y}" loading="lazy"> eine Abbildung, so dass für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> und für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> gilt:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (g\triangleright x)=g\star \varphi (x),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24075f1756cde26ec852ceeeb91f58e6bed231e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.685ex; height:2.843ex;" alt="{\displaystyle \varphi (g\triangleright x)=g\star \varphi (x),}" loading="lazy"></dd></dl>

dann wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> als <a href="%C3%84quivariante_Abbildung" title="Äquivariante Abbildung">G-äquivariant</a> oder auch als Homomorphismus von <a href="G-Menge" class="mw-redirect" title="G-Menge"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Mengen</a> bezeichnet.

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy"> die (Links-)Operation einer Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> auf einer Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> nennt man dann

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\triangleright x:=\{g\triangleright x\mid g\in G\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2a85217247ac28ce7db91ac177e30cddef43712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.782ex; height:2.843ex;" alt="{\displaystyle G\triangleright x:=\{g\triangleright x\mid g\in G\}}" loading="lazy"></dd></dl>

die Bahn, das Transitivitätsgebiet, das Transitivitätssystem oder den Orbit (engl. orbit) von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d07e9f568a88785ae48006ac3c4b951020f1699a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}" loading="lazy">

Die Bahnen bilden eine <a href="Partition_(Mengenlehre)" title="Partition (Mengenlehre)">Partition</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> Die Anzahl der Elemente einer Bahn (bzw. ihre <a href="M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Mächtigkeit</a>) wird auch die Länge der Bahn genannt. Für ein fest gewähltes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b79e955b57dd7aada93b8afd459996ae941d480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}" loading="lazy"> nennt man die durch

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\mapsto g\triangleright x_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9342da096ee74e4d7491f7f5ad89e19f3f42511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.425ex; height:2.176ex;" alt="{\displaystyle g\mapsto g\triangleright x_{0}}" loading="lazy"></dd></dl>

gegebene Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\to X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92ac8b29f3aa93549c3871f007d9186523c967a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.421ex; height:2.176ex;" alt="{\displaystyle G\to X}" loading="lazy"> die „Orbitabbildung“.

Die Bahnen sind die <a href="%C3%84quivalenzklasse" class="mw-redirect" title="Äquivalenzklasse">Äquivalenzklassen</a> bezüglich der Äquivalenzrelation der Konjugation (unter der Operation der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, <a href="Scil." class="mw-redirect" title="Scil.">scil.</a>):

<dl><dd>Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}" loading="lazy"> heißen (unter der Gruppenoperation) konjugiert, (in Zeichen: „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\sim y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbd1014d850b7c883eb76301dd58c643e3c7e4eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\sim y}" loading="lazy">“) genau dann, falls es ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> gibt, für das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c119f9876c0de5c965e76d001f15c492e3faf1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.894ex; height:2.176ex;" alt="{\displaystyle g\triangleright x=y}" loading="lazy"> gilt.<a href="#cite_note-1">[Anm 1]</a></dd></dl>

Offenbar sind zwei Elemente einer Bahn stets konjugiert.

Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\backslash X:=\{G\triangleright x\mid x\in X\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\backslash X:=\{G\triangleright x\mid x\in X\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d004c0b563b61b0fc7ba37f52decffc7848499ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.478ex; height:2.843ex;" alt="{\displaystyle G\backslash X:=\{G\triangleright x\mid x\in X\}}" loading="lazy"> der Äquivalenzklassen wird Bahnenraum oder Orbitraum genannt und besteht demnach aus den Konjugationsklassen (unter der Gruppenoperation).

Für eine Rechtsoperation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleleft }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleleft }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f51c6780e67750600191c48af0a9a2af862c5e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleleft }" loading="lazy"> definiert man analog

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\triangleleft G:=\{x\triangleleft g\mid g\in G\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/366549baa82d015ea53d81fae75783261c37e5e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.782ex; height:2.843ex;" alt="{\displaystyle x\triangleleft G:=\{x\triangleleft g\mid g\in G\}}" loading="lazy"></dd></dl>

und

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X/G:=\{x\triangleleft G\mid x\in X\}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c55a086c1fb628ebdfba09b08a24a4bbd5d9eb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.125ex; height:2.843ex;" alt="{\displaystyle X/G:=\{x\triangleleft G\mid x\in X\}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Fundamentalbereich">Fundamentalbereich</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Fundamentalbereich" title="Fundamentalbereich">Fundamentalbereich</a></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> eine Menge und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eine Transformationsgruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. Für einen <a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> bezeichne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\triangleright x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\triangleright x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df7d09f34e672e7141b74f762c8de942a8d155f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle G\triangleright x}" loading="lazy"> die <a href="#Bahn">Bahn</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. Dann heißt die <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subset X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\subset X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/253df0b206fc9c6fa0e67b94d7eb807e9f253274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.819ex; height:2.176ex;" alt="{\displaystyle F\subset X}" loading="lazy"> ein <a href="Fundamentalbereich" title="Fundamentalbereich">Fundamentalbereich</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, wenn der <a href="Schnittmenge" class="mw-redirect" title="Schnittmenge">Schnitt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\triangleright x\cap F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\triangleright x\cap F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c71e4c7154f41c331427564cb97a3a9e22b30bac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.674ex; height:2.176ex;" alt="{\displaystyle G\triangleright x\cap F}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> eine einelementige Menge ist.<a href="#cite_note-2">[1]</a>

Mit anderen Worten enthält ein Fundamentalbereich genau einen Punkt aus jeder Bahn und ist damit dasselbe wie ein <a href="Repr%C3%A4sentantensystem" class="mw-redirect" title="Repräsentantensystem">Repräsentantensystem</a> der durch die Operation auf der Menge induzierten Äquivalenzrelation.

<dl><dt>Beispiel</dt>

<dd>Das Quadrat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1)\times [0,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1)\times [0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2a5c8feb02f7c84fc137e0db8ae9a67aec97d04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.661ex; height:2.843ex;" alt="{\displaystyle [0,1)\times [0,1)}" loading="lazy"> ist ein Fundamentalbereich von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"> bezüglich der Transformationsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a807ab4cb3de13a66771b5a303aca31e0391e6aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{2}}" loading="lazy">. Jeder Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)\in \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7436cca5d34cde6f59fd9989fe0e996e403acfb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.901ex; height:3.176ex;" alt="{\displaystyle (x,y)\in \mathbb {R} ^{2}}" loading="lazy"> lässt sich schreiben als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u+m,v+n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u+m,v+n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f0440559ba21d6bdf6c6f457d7f88e3b3f3b984.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.416ex; height:2.843ex;" alt="{\displaystyle (u+m,v+n)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,n):=(\lfloor x\rfloor ,\lfloor y\rfloor )\in \mathbb {Z} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,n):=(\lfloor x\rfloor ,\lfloor y\rfloor )\in \mathbb {Z} ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b0ba36ae5a6880db66dd138914988c0c65bf4ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.927ex; height:3.176ex;" alt="{\displaystyle (m,n):=(\lfloor x\rfloor ,\lfloor y\rfloor )\in \mathbb {Z} ^{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v):=(x-m,y-n)\in [0,1)\times [0,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u,v):=(x-m,y-n)\in [0,1)\times [0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4126bee83974a38f6369a59a859a5b63a94fc310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.992ex; height:2.843ex;" alt="{\displaystyle (u,v):=(x-m,y-n)\in [0,1)\times [0,1)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Transitive_und_scharf_transitive_Operationen">Transitive und scharf transitive Operationen </h3></div>

Man bezeichnet die Gruppenoperation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> als (einfach) transitiv oder sagt „die Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> operiert (einfach) transitiv auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">“, wenn es zu je zwei Elementen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> gibt, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c119f9876c0de5c965e76d001f15c492e3faf1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.894ex; height:2.176ex;" alt="{\displaystyle g\triangleright x=y}" loading="lazy"> gilt. Es genügt sogar, dies nur für ein fest gewähltes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'\in X}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x'\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81084d8383a416e5ca37b0ea531a56eebb02661c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.835ex; height:2.509ex;" alt="{\displaystyle x'\in X}" loading="lazy"> zu fordern, das heißt: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy">, so operiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> (einfach) transitiv, wenn es zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6015f751d0278b3aa3b5e4c33740456f08e888b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.976ex; height:2.509ex;" alt="{\displaystyle y\in X}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> gibt mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c119f9876c0de5c965e76d001f15c492e3faf1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.894ex; height:2.176ex;" alt="{\displaystyle g\triangleright x=y}" loading="lazy">. („Jedes Element von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist zu einem fest gewählten konjugiert.“)

Ist ein Gruppenelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c119f9876c0de5c965e76d001f15c492e3faf1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.894ex; height:2.176ex;" alt="{\displaystyle g\triangleright x=y}" loading="lazy"> darüber hinaus durch zwei beliebige Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}" loading="lazy"> eindeutig bestimmt, so nennt man die Gruppenoperation scharf (einfach) transitiv.

Transitive Gruppenoperationen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> sind demnach dadurch gekennzeichnet, dass sie nur eine <a href="#Bahn">Bahn</a> besitzen, die ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> umfasst, mit anderen Worten: Je zwei beliebige Elemente aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> sind zueinander konjugiert.

Besitzt sie mehr als nur eine Bahn, so heißt die Gruppenoperation intransitiv.

Gibt es sogar zu jedem <a href="Paarmenge" title="Paarmenge">Paar</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2})\in X^{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2})\in X^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01053b7e0ea24044bf7811316d76f937a4dc199a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.503ex; height:3.176ex;" alt="{\displaystyle (x_{1},x_{2})\in X^{2}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\neq x_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}\neq x_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae809ae1f500b6f959ee08a5895ed2545ceb6827.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.866ex; height:2.676ex;" alt="{\displaystyle x_{1}\neq x_{2}}" loading="lazy"> und jedem Paar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{1},y_{2})\in X^{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y_{1},y_{2})\in X^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/513f76e0f0c0a042f2a1cd58aa1e0861c363b800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.122ex; height:3.176ex;" alt="{\displaystyle (y_{1},y_{2})\in X^{2}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}\neq y_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{1}\neq y_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31224901735ffd6689f888777908546c40b18067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.485ex; height:2.676ex;" alt="{\displaystyle y_{1}\neq y_{2}}" loading="lazy"> ein Gruppenelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy">, für das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x_{1}=y_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x_{1}=y_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8fb26aa38fb75637c93094f0ebebbb2f727207d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.987ex; height:2.176ex;" alt="{\displaystyle g\triangleright x_{1}=y_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x_{2}=y_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x_{2}=y_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c774586f61637f82477553bcc350aee441064254.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.987ex; height:2.176ex;" alt="{\displaystyle g\triangleright x_{2}=y_{2}}" loading="lazy"> ist, dann nennt man die Gruppenoperation zweifach transitiv und scharf zweifach transitiv, wenn es stets genau ein Gruppenelement mit der genannten Eigenschaft gibt.

Allgemein induziert eine Operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy"> der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} \setminus \{0\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} \setminus \{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/271ff5eb3b6b5065b8496abf4ad8e326984d5dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.412ex; height:2.843ex;" alt="{\displaystyle k\in \mathbb {N} \setminus \{0\}}" loading="lazy"> eine Operation

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright _{k}\colon G\times X^{k}\to X^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40ccf3a0fd1495b414122c67cc22913b8708c5df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.737ex; height:3.009ex;" alt="{\displaystyle \triangleright _{k}\colon G\times X^{k}\to X^{k}}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright _{k}(x_{1},x_{2},\dotsc ,x_{k})=(g\triangleright x_{1},g\triangleright x_{2},\dotsc ,g\triangleright x_{k}).}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright _{k}(x_{1},x_{2},\dotsc ,x_{k})=(g\triangleright x_{1},g\triangleright x_{2},\dotsc ,g\triangleright x_{k}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac10405dbf5eaba576ae29c21c0dcd42600d192a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.492ex; height:2.843ex;" alt="{\displaystyle g\triangleright _{k}(x_{1},x_{2},\dotsc ,x_{k})=(g\triangleright x_{1},g\triangleright x_{2},\dotsc ,g\triangleright x_{k}).}" loading="lazy"></dd></dl>

Da hierbei stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=x_{j}\Leftrightarrow g\triangleright x_{i}=g\triangleright x_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=x_{j}\Leftrightarrow g\triangleright x_{i}=g\triangleright x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/674eaafc9bf1f3e3f5f9d498be9b948b9436dd1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.17ex; height:2.509ex;" alt="{\displaystyle x_{i}=x_{j}\Leftrightarrow g\triangleright x_{i}=g\triangleright x_{j}}" loading="lazy"> gilt, lässt sich diese Operation auf jene Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X^{k})'\subseteq X^{k}}">

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X^{k})'\subseteq X^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f384751be2fce4a5e51bc784da7c1c4f70577c3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.764ex; height:3.176ex;" alt="{\displaystyle (X^{k})'\subseteq X^{k}}" loading="lazy"> beschränken, die nur diejenigen Tupel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{1\leq i\leq k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{i})_{1\leq i\leq k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/158fa36c2b64872857782d6c79827d6a054b97f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.974ex; height:2.843ex;" alt="{\displaystyle (x_{i})_{1\leq i\leq k}}" loading="lazy"> mit paarweise verschiedenen Einträgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> enthält.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright _{k}\colon (X^{k})'\to (X^{k})'}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright _{k}\colon (X^{k})'\to (X^{k})'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9284d17e17d159a3bf0dd0375ddc6f8e35e61281.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.058ex; height:3.176ex;" alt="{\displaystyle \triangleright _{k}\colon (X^{k})'\to (X^{k})'}" loading="lazy"> (scharf) einfach transitiv, dann heißt die Gruppenoperation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy"> (scharf) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach transitiv. Mit anderen Worten: Die Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> operiert via <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy"> genau dann

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach transitiv auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright _{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright _{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f34923ae906b0dbf9965aa6a4f00892943c0efa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.251ex; height:2.176ex;" alt="{\displaystyle \triangleright _{k}}" loading="lazy"> transitiv operiert,</li>

<li>scharf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach transitiv auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright _{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright _{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f34923ae906b0dbf9965aa6a4f00892943c0efa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.251ex; height:2.176ex;" alt="{\displaystyle \triangleright _{k}}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X^{k})'}">

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X^{k})'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e01682c629c660774b251e887906696e2fe431a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.58ex; height:3.176ex;" alt="{\displaystyle (X^{k})'}" loading="lazy"> scharf einfach transitiv operiert, das heißt, wenn es für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},t_{2}\in (X^{k})'}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{1},t_{2}\in (X^{k})'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba18a22a633bd07c31c06388e10b095203983dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.242ex; height:3.176ex;" alt="{\displaystyle t_{1},t_{2}\in (X^{k})'}" loading="lazy"> (also für „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Tupel ohne Doppelnennung“) stets genau ein Gruppenelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright _{k}t_{1}=t_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright _{k}t_{1}=t_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8097ad3e4f9296d1ad431d62ddb5947bf57a4c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.286ex; height:2.343ex;" alt="{\displaystyle g\triangleright _{k}t_{1}=t_{2}}" loading="lazy"> gibt.</li></ul>

Wichtige Anwendungen haben solche (scharf) transitiven Operationen in der <a href="Geometrie" title="Geometrie">Geometrie</a>, siehe zum Beispiel <a href="Affinit%C3%A4t_(Mathematik)" title="Affinität (Mathematik)">Affinität (Mathematik)</a>, <a href="Moufangebene" class="mw-redirect" title="Moufangebene">Moufangebene</a>, <a href="Affine_Translationsebene" title="Affine Translationsebene">Affine Translationsebene</a>.

Bei der Anwendung auf Graphen begegnet man, wenn keine Missverständnisse zu befürchten sind, den kurzen Formulierungen „der Graph ist kantentransitiv“ oder „die Gruppe ist kantentransitiv“ (engl. edge-transitive) anstelle der ausführlichen Formulierung „die Symmetriegruppe des Graphen operiert transitiv auf den Kanten“.

<dl><dt>Beispiele</dt></dl>

<ul><li>Die <a href="Kleinsche_Vierergruppe" title="Kleinsche Vierergruppe">(Kleinsche) Vierergruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}:=\{e,(12)(34),(13)(24),(14)(23)\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{1}:=\{e,(12)(34),(13)(24),(14)(23)\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49c40c6cce5f5a5f5d30b9cde6ee16be40368b6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.47ex; height:2.843ex;" alt="{\displaystyle V_{1}:=\{e,(12)(34),(13)(24),(14)(23)\}}" loading="lazy"> operiert (scharf einfach) transitiv auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\{1,2,3,4\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:=\{1,2,3,4\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c88120d298d5185fcdc5b9b8edaa09f8cb79aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.802ex; height:2.843ex;" alt="{\displaystyle X:=\{1,2,3,4\}}" loading="lazy">, da die Ziffer 1 in jede andere übergeführt werden kann. Dies gilt nicht für die Operation der Vierergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{2}:=\{e,(12),(34),(12)(34)\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{2}:=\{e,(12),(34),(12)(34)\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/967f3bb99606f219632f17e55f6535837dd33b02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.202ex; height:2.843ex;" alt="{\displaystyle V_{2}:=\{e,(12),(34),(12)(34)\}}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. Dabei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adfdbc929f16cb00bb43289c223651b41f7b9f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceaa689a894f5020a7b46177d201cbce2d41122b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{2}}" loading="lazy"> zueinander isomorph und zur <a href="Diedergruppe" title="Diedergruppe">Diedergruppe</a> der Ordnung vier.</li>

<li>Die <a href="Galoisgruppe" title="Galoisgruppe">Galoisgruppe</a> eines über einem Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> <a href="Irreduzibles_Polynom" title="Irreduzibles Polynom">irreduziblen</a> <a href="Polynom" title="Polynom">Polynoms</a> (etwa eines Polynoms mit rationalen Koeffizienten (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0854290031d92bf519f3ed79754edbe3eec1db77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.973ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {Q} }" loading="lazy">)) operiert transitiv auf der Menge der Nullstellen des Polynoms.<a href="#cite_note-3">[2]</a><a href="#cite_note-4">[3]</a> Dieses Beispiel ist Gegenstand des Abschnittes <a href="#Automorphismengruppe_eines_Körpers_und_einer_Galois-Erweiterung">„Automorphismengruppe eines Körpers und einer Galois-Erweiterung“</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Homogene_Operationen">Homogene Operationen</h3></div>

Eine Verallgemeinerung der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach transitiven Operation ist die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach homogene Operation. Eine Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> operiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach homogen auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} \setminus \{0\},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} \setminus \{0\},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45c757eefcedb8a26e69c1ae6d32c97c8b03e9e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.059ex; height:2.843ex;" alt="{\displaystyle k\in \mathbb {N} \setminus \{0\},}" loading="lazy"> wenn es für zwei beliebige Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,V\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U,V\subseteq X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0a785220eca8838b85c93568857c7b35f784119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.682ex; height:2.509ex;" alt="{\displaystyle U,V\subseteq X}" loading="lazy"> mit je genau <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> Elementen stets mindestens ein Gruppenelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> gibt, das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> abbildet, also mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright U=V.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright U=V.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6737a1a346f7f41328c117ce97751d09e132ee6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.626ex; height:2.509ex;" alt="{\displaystyle g\triangleright U=V.}" loading="lazy"> Jede <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach transitive Operation ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach homogen. Von der homogenen Operation wird im Unterschied zur transitiven Operation nicht verlangt, dass die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> vorgegebenen Urbildelemente in einer bestimmten Reihenfolge auf die vorgegebenen Bildelemente abgebildet werden.

<div class="mw-heading mw-heading3"><h3 id="Stabilisator">Stabilisator</h3></div>

Für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> nennt man

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}=\{g\in G\mid g\triangleright x=x\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/416de45bb920916789f7f214e6de8c9cba129d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.212ex; height:2.843ex;" alt="{\displaystyle G_{x}=\{g\in G\mid g\triangleright x=x\}}" loading="lazy"></dd></dl>

den Stabilisator, die Isotropiegruppe, die Fixgruppe oder die Standuntergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.\;(G_{x},*)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x.\;(G_{x},*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe29162dd65b29c3086a8ff153e71c196e4e2689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.014ex; height:2.843ex;" alt="{\displaystyle x.\;(G_{x},*)}" loading="lazy"> ist eine Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy">, die auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> operiert. Durch die Operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy"> ist dann eine kanonische Bijektion zwischen der Menge der Linksnebenklassen nach dem Stabilisator des Elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und dessen Bahn gegeben (siehe unten „Bahnensatz“):

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/G_{x}\;{\stackrel {\cong }{\to }}\;G\triangleright x,\quad g*G_{x}\mapsto g\triangleright x.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/999318a1cf84212d93ecb0831dccdaff5c3d9efb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.521ex; height:3.843ex;" alt="{\displaystyle G/G_{x}\;{\stackrel {\cong }{\to }}\;G\triangleright x,\quad g*G_{x}\mapsto g\triangleright x.}" loading="lazy"></dd></dl>

Gehören <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> derselben Bahn an (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\sim y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbd1014d850b7c883eb76301dd58c643e3c7e4eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\sim y}" loading="lazy">) so sind die zugehörigen Fixgruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x},G_{y}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x},G_{y}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/330de35d1d0fbdc1198fa011a6e7320218afbace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.909ex; height:2.843ex;" alt="{\displaystyle G_{x},G_{y}}" loading="lazy"> im Sinne der Gruppentheorie zueinander konjugierte Untergruppen,<a href="#cite_note-5">[Anm 2]</a> das heißt, mit einem geeigneten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}=g^{-1}*G_{y}*g}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}=g^{-1}*G_{y}*g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa1c65a5316aea237f92e2ca7e4bcd0dd8601eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.93ex; height:3.343ex;" alt="{\displaystyle G_{x}=g^{-1}*G_{y}*g}" loading="lazy">. Dazu wähle nämlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=g\triangleright x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=g\triangleright x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fa15e9ed07d3c58e0901aba6222e861dfe3146d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.894ex; height:2.176ex;" alt="{\displaystyle y=g\triangleright x}" loading="lazy"> gemäß Voraussetzung. Kurz: Fixgruppen konjugierter Elemente sind konjugierte Untergruppen.

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2f78101676239fc50bea86722dedf1512180e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle G_{x}}" loading="lazy"> operiert (durch Einschränkung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy">) auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\setminus \{x\}.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\setminus \{x\}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/860b9808cbf45921ae79e40a6eb87249eb3deaa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.476ex; height:2.843ex;" alt="{\displaystyle X\setminus \{x\}.}" loading="lazy"> Ist diese Operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fach transitiv und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}\neq G,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}\neq G,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ff2676a1b734e757e406b1ef482f1107fffc750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.571ex; height:2.676ex;" alt="{\displaystyle G_{x}\neq G,}" loading="lazy"> so ist die Operation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> sogar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f9f13644a6be482d7ddb19a6e0c706564773085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+1)}" loading="lazy">-fach transitiv.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\subseteq X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abf3e597919d6ae699180ce66b51edbe14f3546a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle Y\subseteq X}" loading="lazy"> eine Teilmenge und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subseteq G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\subseteq G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dbe10924c5c9134471b96d6a8dfb9c8b1bf0f04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.989ex; height:2.343ex;" alt="{\displaystyle H\subseteq G}" loading="lazy"> eine Untergruppe, und gilt

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\triangleright Y\subseteq Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c01a25af2f7fc37dfdc9eda4bd303ba033c8d6b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.903ex; height:2.343ex;" alt="{\displaystyle H\triangleright Y\subseteq Y}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\triangleright Y:=\{h\triangleright y\mid h\in H,y\in Y\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\triangleright Y:=\{h\triangleright y\mid h\in H,y\in Y\},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4016be165362c30a7a32b15ed228027e200bff24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.422ex; height:2.843ex;" alt="{\displaystyle H\triangleright Y:=\{h\triangleright y\mid h\in H,y\in Y\},}" loading="lazy"></dd></dl>

so sagt man, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> stabil oder invariant unter (der Operation von) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> ist oder dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> stabilisiert wird. Es gilt dann stets sogar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\triangleright Y=Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\triangleright Y=Y.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86d694a0017d874b9cd115ec8d53dd8769df5780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.55ex; height:2.176ex;" alt="{\displaystyle H\triangleright Y=Y.}" loading="lazy"> Der Stabilisator eines Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> ist also die maximale Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy"> stabilisiert.

Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\supset Y'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\supset Y'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf803d8a50b27d2e37ea79560e0f7ac584c03702.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.457ex; height:2.509ex;" alt="{\displaystyle Y\supset Y'}" loading="lazy">, so ist die maximale Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> stabilisiert, eine Teilmenge der maximalen Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bacdb2a4ddcc06c2dfbb7af3d92aec37bbaaf0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.585ex; height:2.343ex;" alt="{\displaystyle Y'}" loading="lazy"> stabilisiert.

<div class="mw-heading mw-heading3"><h3 id="Fixgruppen_von_Mengen_und_ihre_Normalisatoren">Fixgruppen von Mengen und ihre Normalisatoren</h3></div>

Für eine (nicht notwendig echte) Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\subseteq X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abf3e597919d6ae699180ce66b51edbe14f3546a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle Y\subseteq X}" loading="lazy"> bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{Y}:=\operatorname {Fix} _{G}(Y):=\{g\in G\mid \forall y\in Y:g\triangleright y=y\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{Y}:=\operatorname {Fix} _{G}(Y):=\{g\in G\mid \forall y\in Y:g\triangleright y=y\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6287ef6d0f5a59f3a9f30bca9e2176bf3ac482c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.067ex; height:2.843ex;" alt="{\displaystyle G_{Y}:=\operatorname {Fix} _{G}(Y):=\{g\in G\mid \forall y\in Y:g\triangleright y=y\}}" loading="lazy"> die Menge aller Gruppenelemente, die jeden Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy"> unverändert lassen.

Die Fixgruppe von einer Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> ist in der maximalen die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> stabilisierenden Untergruppe<a href="#cite_note-6">[Anm 3]</a> enthalten, jedoch im Allgemeinen nicht mit ihr identisch, da diese die Punkte aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> nicht punktweise fixieren muss, sondern permutieren darf.

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c938e8bf0bc7fa38baf00c88fa8aa6acf633f823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle h\in G}" loading="lazy"> folgt unmittelbar aus der Definition:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} _{G}(h\triangleright Y)=h*\operatorname {Fix} _{G}(Y)*h^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb1b31b122e41e152feee8fb0f477017b2453d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.03ex; height:3.176ex;" alt="{\displaystyle \operatorname {Fix} _{G}(h\triangleright Y)=h*\operatorname {Fix} _{G}(Y)*h^{-1}}" loading="lazy">.</li>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y':=h\triangleright Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec15785c2504d6dd0680f73dec53372555ee0258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.637ex; height:2.509ex;" alt="{\displaystyle Y':=h\triangleright Y}" loading="lazy"> und bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\{g\in G\mid g\triangleright Y=Y\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=\{g\in G\mid g\triangleright Y=Y\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/692bbb06687eb7ba6c56c875b763fc61b15873ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.882ex; height:2.843ex;" alt="{\displaystyle U=\{g\in G\mid g\triangleright Y=Y\}}" loading="lazy"> die maximale die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> stabilisierende Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecaad8b1d7c1c60a880b7221d6bb70337b83b5aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.526ex; height:2.509ex;" alt="{\displaystyle U'}" loading="lazy"> die Entsprechung für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bacdb2a4ddcc06c2dfbb7af3d92aec37bbaaf0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.585ex; height:2.343ex;" alt="{\displaystyle Y'}" loading="lazy">, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U'=h^{-1}*U*h}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U'=h^{-1}*U*h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c06375d313c5db52d6e1be4f1636a565ae4e1fb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.807ex; height:2.676ex;" alt="{\displaystyle U'=h^{-1}*U*h}" loading="lazy">, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in U'\Leftrightarrow (g*h)\triangleright Y=h\triangleright Y\Leftrightarrow h^{-1}*g*h\in U}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in U'\Leftrightarrow (g*h)\triangleright Y=h\triangleright Y\Leftrightarrow h^{-1}*g*h\in U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/507f490ae71f509e8e0d725eee1545de5893a59b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.683ex; height:3.176ex;" alt="{\displaystyle g\in U'\Leftrightarrow (g*h)\triangleright Y=h\triangleright Y\Leftrightarrow h^{-1}*g*h\in U}" loading="lazy">.</li></ul>

Zusammenfassend: Die maximal stabilisierenden Untergruppen (bzw. die Fixgruppen) konjugierter Teilmengen sind unter der Operation der inneren Automorphismen (siehe <a href="#Operation_durch_Konjugation">Beispiel „Operation durch Konjugation“</a>) konjugiert. Insbesondere sind die maximal stabilisierenden Untergruppen (bzw. die Fixgruppen) von Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, die unter der Operation der gesamten Transformationsgruppe stabil (invariant) sind <a href="Normalteiler" title="Normalteiler">Normalteiler</a> in der Transformationsgruppe.

Auch wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> nicht invariant unter der gesamten Transformationsgruppe ist, so folgt aus der Identität

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} _{G}(h\triangleright Y)=h*\operatorname {Fix} _{G}(Y)*h^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb1b31b122e41e152feee8fb0f477017b2453d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.03ex; height:3.176ex;" alt="{\displaystyle \operatorname {Fix} _{G}(h\triangleright Y)=h*\operatorname {Fix} _{G}(Y)*h^{-1}}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c938e8bf0bc7fa38baf00c88fa8aa6acf633f823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle h\in G}" loading="lazy"> (und unter Beachtung der Tatsache <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fix} _{G}(Y)\subseteq U}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} _{G}(Y)\subseteq U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75caec25aff7de334540bdada93a1d1d150bf9b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.38ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fix} _{G}(Y)\subseteq U}" loading="lazy">) die Äquivalenz

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in U\Leftrightarrow h*\operatorname {Fix} _{G}(Y)*h^{-1}=\operatorname {Fix} _{G}(Y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3be85a91bb4af4cbca6cb89b77d99053f65bcfa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.072ex; height:3.176ex;" alt="{\displaystyle h\in U\Leftrightarrow h*\operatorname {Fix} _{G}(Y)*h^{-1}=\operatorname {Fix} _{G}(Y)}" loading="lazy">.</dd></dl>

Also stehen die Fixgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fix} _{G}(Y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} _{G}(Y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bd93505d88e82a0ef61723851b2163517f85f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.499ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fix} _{G}(Y)}" loading="lazy"> und die sie umfassende maximale Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, unter deren Operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> invariant ist, in dem bemerkenswerten Verhältnis:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=\{g\in G\mid g*\operatorname {Fix} _{G}(Y)*g^{-1}=\operatorname {Fix} _{G}(Y)\}=:\operatorname {Nor} _{G}(\operatorname {Fix} _{G}(Y))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6460ebce84d020691ff3cc6b63b216ee71cddf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.373ex; height:3.176ex;" alt="{\displaystyle U=\{g\in G\mid g*\operatorname {Fix} _{G}(Y)*g^{-1}=\operatorname {Fix} _{G}(Y)\}=:\operatorname {Nor} _{G}(\operatorname {Fix} _{G}(Y))}" loading="lazy">. Die rechte Menge heißt der <a href="Normalisator" title="Normalisator">Normalisator</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fix} _{G}(Y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} _{G}(Y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bd93505d88e82a0ef61723851b2163517f85f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.499ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fix} _{G}(Y)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, da sie die maximale Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ist, in der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fix} _{G}(Y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} _{G}(Y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bd93505d88e82a0ef61723851b2163517f85f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.499ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fix} _{G}(Y)}" loading="lazy"> als Normalteiler enthalten ist. Diese ist identisch mit der maximalen Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Inv} _{G}(Y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Inv} _{G}(Y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/780fddf0f0e8fa6e3cb1088586404ba83ad36b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.466ex; height:2.843ex;" alt="{\displaystyle \operatorname {Inv} _{G}(Y)}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, unter deren Operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> invariant (stabil) ist: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Nor} _{G}(\operatorname {Fix} _{G}(Y))=\operatorname {Inv} _{G}(Y):=\{g\in G\mid g\triangleright Y\subseteq Y\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Nor} _{G}(\operatorname {Fix} _{G}(Y))=\operatorname {Inv} _{G}(Y):=\{g\in G\mid g\triangleright Y\subseteq Y\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1baf2686e82bcec299bbdc27945a598ea1c67643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.96ex; height:2.843ex;" alt="{\displaystyle \operatorname {Nor} _{G}(\operatorname {Fix} _{G}(Y))=\operatorname {Inv} _{G}(Y):=\{g\in G\mid g\triangleright Y\subseteq Y\}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Fixpunktmengen">Fixpunktmengen</h3></div>

Ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> heißt Fixpunkt unter der Operation eines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x=x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x=x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbd8424a6d2aba417eeb5738a3bf82ad1e5911f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.069ex; height:2.176ex;" alt="{\displaystyle g\triangleright x=x}" loading="lazy">. Es heißt Fixpunkt unter der Gruppe(noperation einer Untergruppe) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subset G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\subset G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15eb570a68da46b0458ff0ead693a82467ab8ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.989ex; height:2.176ex;" alt="{\displaystyle H\subset G}" loading="lazy">, wenn es Fixpunkt eines jeden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/675a79e26028d91d97f4e2ce279c314b0f194c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.243ex; height:2.176ex;" alt="{\displaystyle h\in H}" loading="lazy"> ist.

Für Fixpunktmengen finden sich in der Literatur folgende Schreibweisen:

<ul><li>die Menge aller Fixpunkte unter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">:</li></ul>

<msup>

</mrow>

</msup>

<mtext> ist Fixpunkt unter </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{g}:=\{x\in X\mid x{\text{ ist Fixpunkt unter }}g\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/976c9f655a49a4c4c3bb23145a0aa12211493d10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.248ex; height:2.843ex;" alt="{\displaystyle X^{g}:=\{x\in X\mid x{\text{ ist Fixpunkt unter }}g\}}" loading="lazy"></dd></dl></dd></dl>

<ul><li>die Menge aller Fixpunkte unter einer Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subset G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\subset G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15eb570a68da46b0458ff0ead693a82467ab8ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.989ex; height:2.176ex;" alt="{\displaystyle H\subset G}" loading="lazy">:</li></ul>

<msup>

</mrow>

</msup>

<mtext> ist Fixpunkt unter der Operation von </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{H}:=\{x\in X\mid h{\text{ ist Fixpunkt unter der Operation von }}H\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a165a3c40b9a8cae827a456a17ffa62db61ae341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.749ex; height:3.176ex;" alt="{\displaystyle X^{H}:=\{x\in X\mid h{\text{ ist Fixpunkt unter der Operation von }}H\}}" loading="lazy"></dd></dl></dd></dl>

Erzeugt das Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c938e8bf0bc7fa38baf00c88fa8aa6acf633f823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle h\in G}" loading="lazy"> die Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{h}=X^{H}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{h}=X^{H}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e9da1850305cc9674e9222229901d2519b5ba6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.963ex; height:2.676ex;" alt="{\displaystyle X^{h}=X^{H}}" loading="lazy">.

Selbstverständlich ist die Fixpunktmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{H}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{H}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87a5069ab37d35fd116be4afe3e2a70db52e1a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.688ex; height:2.676ex;" alt="{\displaystyle X^{H}}" loading="lazy"> stabil unter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">, doch ist es im Allgemeinen nicht gleich der maximalen unter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> stabilen Teilmenge, denn in dieser müssen die Elemente nicht punktweise fix bleiben, sondern dürfen untereinander permutiert werden.

Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{H}\subset X^{H'}}">

<msup>

</mrow>

</msup>

<msup>

<msup>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{H}\subset X^{H'}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9e75338ce1ffda25e511b725c8a5e23d9d546a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.035ex; height:2.843ex;" alt="{\displaystyle X^{H}\subset X^{H'}}" loading="lazy">, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\supset H'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\supset H'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6ff0180a098a0a1d17a0a79a0ea5fcfe96d5a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.95ex; height:2.509ex;" alt="{\displaystyle H\supset H'}" loading="lazy">, wie leicht zu erkennen.

Des Weiteren gilt offensichtlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fix} _{G}(X^{H})\supset H}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} _{G}(X^{H})\supset H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e8afa141de6438b6fca3261ff497e933c8b0ced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.576ex; height:3.176ex;" alt="{\displaystyle \operatorname {Fix} _{G}(X^{H})\supset H}" loading="lazy">. Der Übergang von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\mapsto \operatorname {Fix} _{G}(X^{H})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\mapsto \operatorname {Fix} _{G}(X^{H})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c706708d26754262264ccdf02e20f0563fa6c28a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.091ex; height:3.176ex;" alt="{\displaystyle H\mapsto \operatorname {Fix} _{G}(X^{H})}" loading="lazy"> wird gelegentlich als „Abschluss der Untergruppe“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> (im Sinne der Gruppenoperation) bezeichnet.<a href="#cite_note-7">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Konjugierte_Mengen">Konjugierte Mengen</h3></div>

In Verallgemeinerung des obigen Begriffs der Konjugation nennt man zwei Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{1},Y_{2}\subset X}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{1},Y_{2}\subset X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54ea4438b99b314165e2a1ab216f17afda15e106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.922ex; height:2.509ex;" alt="{\displaystyle Y_{1},Y_{2}\subset X}" loading="lazy"> (unter der Gruppenoperation) konjugiert, wenn es ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> mit der Eigenschaft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright Y_{1}=Y_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright Y_{1}=Y_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea2b0ab75a2d30254c6482ff5ef0f6d142fb57ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.219ex; height:2.509ex;" alt="{\displaystyle g\triangleright Y_{1}=Y_{2}}" loading="lazy"> gibt.

Schränkt man die transitive Operation einer Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf einen Normalteiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> ein, so sind deren Bahnen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\triangleright x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\triangleright x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c87b2039837ec7a337c4e6e82420b3bb4af11cf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.588ex; height:2.176ex;" alt="{\displaystyle N\triangleright x}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x\in X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x\in X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c0f14c8a02ef99de1ecd41696b1a22c17248f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.96ex; height:2.843ex;" alt="{\displaystyle (x\in X)}" loading="lazy"> sämtlich unter der Gruppenoperation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> zueinander konjugiert, das heißt: Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}" loading="lazy">, so gibt es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright (N\triangleright x)=N\triangleright y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright (N\triangleright x)=N\triangleright y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43dd29a8e8e33dc94a7229bcc9af1cdabc04f0a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.22ex; height:2.843ex;" alt="{\displaystyle g\triangleright (N\triangleright x)=N\triangleright y}" loading="lazy">. Dazu wähle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright x=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright x=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c119f9876c0de5c965e76d001f15c492e3faf1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.894ex; height:2.176ex;" alt="{\displaystyle g\triangleright x=y}" loading="lazy"> und beachte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*N=N*g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*N=N*g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46471ed8b618ebc2cd72badeeafb79d20d7fe373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.847ex; height:2.509ex;" alt="{\displaystyle g*N=N*g}" loading="lazy"> gemäß Voraussetzung.<a href="#cite_note-8">[5]</a>

Auf den Transitivitätsgebieten unter einem Normalteiler operiert die Quotientengruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab52bff253c690c4e0d473400ab8c365ea019298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}" loading="lazy"> wohldefiniert.

Ferner folgt: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> endlich, so besitzen die Transitivitätsgebiete unter einem Normalteiler dieselbe Mächtigkeit.

<div class="mw-heading mw-heading3"><h3 id="Primitive_Gruppenoperationen_und_Imprimitivitätsgebiete">Primitive Gruppenoperationen und Imprimitivitätsgebiete</h3></div>

Für transitive Gruppenoperationen gibt es zwei triviale Partitionierungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\biguplus _{i\in I}Y_{i}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\biguplus _{i\in I}Y_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3d3fdd5f9be2469f087fd7a715c5e5e3512d496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.197ex; height:5.676ex;" alt="{\displaystyle X=\biguplus _{i\in I}Y_{i}}" loading="lazy"> der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}" loading="lazy">, so dass die Gruppenoperation diese Teilmengen permutiert. Dabei sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> eine nichtleere Indexmenge, die das neutrale Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc993b87bf6f6f7f8e5f1e99011f92bd4b0188a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.176ex;" alt="{\displaystyle e\in G}" loading="lazy"> enthalte. Das die Teilmengen permutiert werden, bedeutet, dass es zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d740fe587228ce31b71c9628e089d1a9b37c6be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}" loading="lazy"> gibt mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright Y_{e}=Y_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright Y_{e}=Y_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87092c5c53e50e8aca992cafb7aaf035e9c92c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.908ex; height:2.509ex;" alt="{\displaystyle g\triangleright Y_{e}=Y_{i}}" loading="lazy">. Daher lässt sich als Indexmenge eine Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\subseteq G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I\subseteq G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bc29cc376828a933690ed54d4bfd03486ecc48d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.097ex; height:2.343ex;" alt="{\displaystyle I\subseteq G}" loading="lazy"> (oder eine gleichmächtige Menge) wählen, die (ohne Einschränkung) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{G}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f65007183ece88b79cbbb5eefcf68576c76a4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.009ex;" alt="{\displaystyle e_{G}}" loading="lazy"> enthalte. Die beiden erwähnten trivialen Partitionierungen sind:

<ul><li>die Partition in einelementige Teilmengen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\biguplus _{x\in X}\{x\}}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\biguplus _{x\in X}\{x\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a86d5462dda3cb443d9254ab3d2794c71852f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.17ex; height:5.676ex;" alt="{\displaystyle X=\biguplus _{x\in X}\{x\}}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ebf998509b76ac8aba39489020132b9ab3d654e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.25ex; height:2.176ex;" alt="{\displaystyle I=X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{x}=\{x\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{x}=\{x\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c726486180f64d3033079ac78e4a53501f4e3e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.276ex; height:2.843ex;" alt="{\displaystyle Y_{x}=\{x\}}" loading="lazy">)<a href="#cite_note-9">[Anm 4]</a> und</li>

<li>die „Partition“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0b549c2e838e7da9dfd27fe4e1eac711331bdef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle X=Y}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\{e\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=\{e\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04f8f329fadd35b58bdf82cc82d7411dd0740499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.679ex; height:2.843ex;" alt="{\displaystyle I=\{e\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{e}=X}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{e}=X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd71a41b37b419869361f89ef23f437a622f668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.427ex; height:2.509ex;" alt="{\displaystyle Y_{e}=X}" loading="lazy">).</li></ul>

Wenn eine transitiv operierende Gruppe nur diese beiden trivialen Partitionierungen besitzt, so heißt ihre Gruppenoperation primitiv, andernfalls imprimitiv.

Operiert sie imprimitiv, so gibt es demnach Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}\subsetneqq X\;(i\in I\subseteq G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{i}\subsetneqq X\;(i\in I\subseteq G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cff3305b32b09f63bec2fcf5027e96366ed45d74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.423ex; height:3.176ex;" alt="{\displaystyle Y_{i}\subsetneqq X\;(i\in I\subseteq G)}" loading="lazy">, die sämtlich mehr als nur ein Element enthalten und eine Partition von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> darstellen. Es lässt sich folgern, dass sie zueinander paarweise konjugiert und infolgedessen gleichmächtig sind.

Die Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}" loading="lazy"> einer nicht-trivialen Partitionierung heißen Imprimitivitätsgebiete.<a href="#cite_note-10">[6]</a>

Die Eigenschaft, primitiv zu operieren, liegt allein in der Gruppe beschlossen, denn für transitiv operierende Gruppen gilt folgendes

<dl><dt>Kriterium für primitive Gruppenoperation</dt>

<dd>Bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2f78101676239fc50bea86722dedf1512180e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle G_{x}}" loading="lazy"> die Fixgruppe eines Elementes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy">, so sind äquivalent:

<ul><li>Es gibt eine echte Zwischengruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}\subsetneqq H\subsetneqq G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}\subsetneqq H\subsetneqq G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9585b508249b3e5f0db98fa839edb91429e19dd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.086ex; height:3.176ex;" alt="{\displaystyle G_{x}\subsetneqq H\subsetneqq G}" loading="lazy">.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> operiert imprimitiv.</li></ul></dd>

<dd>Andernfalls operiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> primitiv, das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist eine primitive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Menge.</dd></dl>

Zum Beweis:

<ul><li>Ist die Partition gegeben, so liege dabei (ohne Beschränkung der Allgemeinheit) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in Y_{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in Y_{e}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77abe95bb6d7fe3ca9264c03bdd5dff2fae82ecb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.519ex; height:2.509ex;" alt="{\displaystyle x\in Y_{e}}" loading="lazy">, und es ist offenbar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> gerade die maximale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{e}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7532a61ca983891fc302b980638b0ff09ab97be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.349ex; height:2.509ex;" alt="{\displaystyle Y_{e}}" loading="lazy"> stabilisierende Untergruppe, das heißt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\{h\in G\mid h\triangleright Y_{e}\subseteq Y_{e}\}=\operatorname {Inv} _{G}(Y_{e})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=\{h\in G\mid h\triangleright Y_{e}\subseteq Y_{e}\}=\operatorname {Inv} _{G}(Y_{e})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/438d07deaf26d30b36511728e24970ff658cdb2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.901ex; height:2.843ex;" alt="{\displaystyle H=\{h\in G\mid h\triangleright Y_{e}\subseteq Y_{e}\}=\operatorname {Inv} _{G}(Y_{e})}" loading="lazy">. Damit gilt notwendig <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}\subsetneqq H\subsetneqq G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}\subsetneqq H\subsetneqq G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9585b508249b3e5f0db98fa839edb91429e19dd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.086ex; height:3.176ex;" alt="{\displaystyle G_{x}\subsetneqq H\subsetneqq G}" loading="lazy">.</li>

<li>Ist umgekehrt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> gegeben so setze man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{e}=H\triangleright x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{e}=H\triangleright x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15ea18fe4ae35a46337ea4d2928a7b1e5f83808b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.035ex; height:2.509ex;" alt="{\displaystyle Y_{e}=H\triangleright x}" loading="lazy"> und folgere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}\subsetneqq Y_{e}\subsetneqq X}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\}\subsetneqq Y_{e}\subsetneqq X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad0a2f528302b2cb59032761499767f24da9fec4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.181ex; height:3.176ex;" alt="{\displaystyle \{x\}\subsetneqq Y_{e}\subsetneqq X}" loading="lazy">. Die Partition in Imprimitivitätsgebiete ist gegeben durch</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{g}:=g\triangleright Y_{e}=(g*H)\triangleright x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c7312f12a042b9191464243be4c9a774900c6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.583ex; height:3.009ex;" alt="{\displaystyle Y_{g}:=g\triangleright Y_{e}=(g*H)\triangleright x}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e205907f4c401e0de8a75ea1b3dd76b14e5bdc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.374ex; height:2.509ex;" alt="{\displaystyle g*H}" loading="lazy"> die Linksnebenklassen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> durchläuft, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> selbst also nur ein geeignetes <a href="Repr%C3%A4sentantensystem" class="mw-redirect" title="Repräsentantensystem">Repräsentantensystem</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\subseteq G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I\subseteq G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bc29cc376828a933690ed54d4bfd03486ecc48d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.097ex; height:2.343ex;" alt="{\displaystyle I\subseteq G}" loading="lazy"> dieser Linksnebenklassen. Insgesamt erhält man die Partition:

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\biguplus _{g*H\in G/H}g\triangleright Y_{e}=\biguplus _{g\in I}\underbrace {(g*H)\triangleright x} _{=Y_{g}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05722fd2125f13bfa34aa683e9ac3b82b566264f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; margin-right: -0.028ex; width:35.8ex; height:7.343ex;" alt="{\displaystyle X=\biguplus _{g*H\in G/H}g\triangleright Y_{e}=\biguplus _{g\in I}\underbrace {(g*H)\triangleright x} _{=Y_{g}}}" loading="lazy">.</dd></dl></dd></dl>

Umgekehrt gilt also: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> operiert genau dann primitiv, wenn die Fixgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2f78101676239fc50bea86722dedf1512180e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle G_{x}}" loading="lazy"> eines Elementes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> eine maximale Untergruppe in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ist.

Die Eigenschaft der Primitivität lässt sich mit Hilfe des Begriffs der <a href="Kongruenzrelation" title="Kongruenzrelation">Kongruenzrelation</a> konziser beschreiben: Eine Äquivalenzrelation „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sim }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }" loading="lazy">“ auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> heiße Kongruenzrelation auf der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, wenn sie mit der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Operation in folgendem Sinne verträglich ist: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y\Rightarrow g\triangleright x\sim g\triangleright y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\sim y\Rightarrow g\triangleright x\sim g\triangleright y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c33ba838cc1e9ff635d6abf82ce29801a22c5f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.403ex; height:2.176ex;" alt="{\displaystyle x\sim y\Rightarrow g\triangleright x\sim g\triangleright y}" loading="lazy">. Genau dann operiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf den Äquivalenzklassen wohldefiniert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright [x]_{\sim }:=[g\triangleright x]_{\sim }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright [x]_{\sim }:=[g\triangleright x]_{\sim }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5675b3c268805090d8067ed371de032b956d0b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.635ex; height:2.843ex;" alt="{\displaystyle g\triangleright [x]_{\sim }:=[g\triangleright x]_{\sim }}" loading="lazy"> („Quotientenbildung nach der Kongruenzrelation“). Die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (bzw. die Operation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">) ist primitiv, wenn sie nur die trivialen Kongruenzrelationen zulässt, nämlich die <a href="Kongruenzrelation#Identitätsrelation" title="Kongruenzrelation">Identität</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]=\{x\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]=\{x\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91348b3024af641c83d5f56224d9af15e5252c8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.376ex; height:2.843ex;" alt="{\displaystyle [x]=\{x\}}" loading="lazy">) und die <a href="Kongruenzrelation#Identitätsrelation" title="Kongruenzrelation">Allrelation (Universalrelation)</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]=X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]=X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e995ef4f2a591a8887fd36f678f80acf7a3e014c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.702ex; height:2.843ex;" alt="{\displaystyle [x]=X}" loading="lazy">).

<div class="mw-heading mw-heading3"><h3 id="Freie_und_treue_Operationen">Freie und treue Operationen</h3></div>

Die Operation heißt frei oder <a href="Permutationsgruppe#Semiregulär_und_regulär" title="Permutationsgruppe">semiregulär</a>, falls jedes Element der Menge nur vom neutralen Element der Gruppe fixiert wird. Das bedeutet, dass sämtliche Stabilisatoren trivial sind, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}=\{e\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}=\{e\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfd775e23efcadbad3f79387282c1a9b460a0ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.506ex; height:2.843ex;" alt="{\displaystyle G_{x}=\{e\}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy">, oder – angesichts des Bahnensatzes äquivalent – dass jede Bahn die Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G:1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f97adaa7944d8f59526ca587d5cb119ab98756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.736ex; height:2.843ex;" alt="{\displaystyle (G:1)}" loading="lazy"> hat. Die Bahnengleichung für endlicher Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> lautet dann vereinfacht:

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \#X=\sum _{x\in R}(G:1)=\#R\cdot (G:1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f4cb9d407beb2bb0ff221bafcabdc30fdeae624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.318ex; height:5.676ex;" alt="{\displaystyle \#X=\sum _{x\in R}(G:1)=\#R\cdot (G:1)}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> ein Repräsentantensystem der Äquivalenzklassen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> bezeichne.</dd></dl>

Die Operation heißt treu bzw. effektiv, falls nur das neutrale Element der Gruppe alle Elemente der Menge zugleich fixiert. Das bedeutet, dass der zugehörige Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }\colon G\to \operatorname {Sym} (X)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }\colon G\to \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/393d619100906bb45e79aa8e7785a36b8623947e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.582ex; height:2.843ex;" alt="{\displaystyle \Theta _{\triangleright }\colon G\to \operatorname {Sym} (X)}" loading="lazy"> trivialen <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> hat, also <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektiv</a> ist.

Jede freie Gruppenoperation auf einer nichtleeren Menge ist treu.

Operiert eine Gruppe nicht treu, so induziert sie eine treue Operation durch die Faktorgruppe nach dem Kern: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\Theta _{\triangleright }}}\colon G/(\ker \Theta _{\triangleright })\to \operatorname {Sym} (X)}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\Theta _{\triangleright }}}\colon G/(\ker \Theta _{\triangleright })\to \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/393104b7f94ef67c1b9fbf5311b2086582384e4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.09ex; height:3.509ex;" alt="{\displaystyle {\overline {\Theta _{\triangleright }}}\colon G/(\ker \Theta _{\triangleright })\to \operatorname {Sym} (X)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Darstellung_der_Transformationsgruppe_durch_Permutationen_bzw._als_Permutationsgruppe">Darstellung der Transformationsgruppe durch Permutationen bzw. als Permutationsgruppe</h3></div>

Bei treuer Operation auf einer endlichen Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> kann die Transformationsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> demnach als Untergruppe der <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrischen Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy"> aufgefasst werden: Man sagt dann, dass „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> als <a href="Permutationsgruppe" title="Permutationsgruppe">Permutationsgruppe</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> operiere“ oder dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a88121f04a342077282bd71e1cfda9c6727c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \Theta _{\triangleright }}" loading="lazy"> „die Transformationsgruppe als Permutationsgruppe darstelle“, nämlich als die Permutationsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }(G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c156d77e6681a9f3cd2d87e450ae53aedd61b27d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.498ex; height:2.843ex;" alt="{\displaystyle \Theta _{\triangleright }(G)}" loading="lazy">. Entsprechend heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a88121f04a342077282bd71e1cfda9c6727c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \Theta _{\triangleright }}" loading="lazy"> eine Permutationsgruppendarstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

Umgekehrt induziert eine Einbettung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\hookrightarrow \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\hookrightarrow \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0934dd2c1553726ff827c0cda75872cd4e48a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.979ex; height:2.843ex;" alt="{\displaystyle G\hookrightarrow \operatorname {Sym} (X)}" loading="lazy"> eine Operation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

In der Sprache der <a href="Darstellungstheorie" title="Darstellungstheorie">Darstellungstheorie</a>, etwa der <a href="Darstellungstheorie_endlicher_Gruppen" title="Darstellungstheorie endlicher Gruppen">endlicher</a> oder <a href="Darstellungstheorie_kompakter_Gruppen" title="Darstellungstheorie kompakter Gruppen">kompakter</a> Gruppen, ist es üblich, von einer Darstellung nicht <a href="A_priori" title="A priori">a priori</a> die <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">Injektivität</a> zu verlangen: Es genügt, dass sie eine homomorphe Abbildung ist. Wenn sie darüber injektiv ist, so heißt sie eine treue Darstellung. Jede Gruppenoperation (auf einer endlichen Menge) liefert eine <a href="Permutationsdarstellung" class="mw-redirect" title="Permutationsdarstellung">Permutationsdarstellung</a>, das heißt eine Darstellung durch Permutationen. Jede treue Permutationsdarstellung ist eine Darstellung als Permutationsgruppe, das heißt eine Permutationsgruppendarstellung, und gibt also eine Interpretation der Transformationsgruppe als eine Gruppe von Permutationen auf einer endlichen Menge. Die Anzahl ihrer Elemente heißt Grad der Darstellung bzw. Operation. Diese Sprechweise befindet sich im Einklang mit den Gepflogenheiten der Darstellungstheorie.<a href="#cite_note-11">[7]</a>

<div class="mw-heading mw-heading4"><h4 id="Äquivalente_Permutationsgruppen">Äquivalente Permutationsgruppen</h4></div>

Zwei Permutationsgruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}\subseteq \operatorname {Sym} (X_{1}),\Pi _{2}\subseteq \operatorname {Sym} (X_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi _{1}\subseteq \operatorname {Sym} (X_{1}),\Pi _{2}\subseteq \operatorname {Sym} (X_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a42fba564713acc0bf13638c5a628aaa90a2c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.313ex; height:2.843ex;" alt="{\displaystyle \Pi _{1}\subseteq \operatorname {Sym} (X_{1}),\Pi _{2}\subseteq \operatorname {Sym} (X_{2})}" loading="lazy"> heißen äquivalent, wenn sie sich nur in der Benennung ihrer Permutationselemente aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d6099d6fb3c34ad5e22fad9c79c40c4ebfee1ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.991ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2}}" loading="lazy"> unterscheiden, soll heißen, wenn es eine Bijektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \colon X_{1}\to X_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi \colon X_{1}\to X_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72333cc1656f706fb3ddff74724e5dacbc0dd2c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.283ex; height:2.509ex;" alt="{\displaystyle \Phi \colon X_{1}\to X_{2}}" loading="lazy"> gibt, die folgendermaßen einen Isomorphismus der Permutationsgruppen stiftet:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\Pi _{1}&\longrightarrow &\Pi _{2}\\\pi &\mapsto &\Phi \circ \pi \circ \Phi ^{-1}\end{matrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0eb6c5d8ad70f040f13818c6243134858d2e9ede.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.41ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}\Pi _{1}&\longrightarrow &\Pi _{2}\\\pi &\mapsto &\Phi \circ \pi \circ \Phi ^{-1}\end{matrix}}}" loading="lazy"></dd></dl>

Ist hierbei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=X_{2}=X}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1}=X_{2}=X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64b68d46539eefa70480bb415b45fe10de032b8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.134ex; height:2.509ex;" alt="{\displaystyle X_{1}=X_{2}=X}" loading="lazy">, so ist natürlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \in \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi \in \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617bed41b08ae86d8f9f0738734aebd8be079a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.764ex; height:2.843ex;" alt="{\displaystyle \Phi \in \operatorname {Sym} (X)}" loading="lazy">, und die Äquivalenz der Permutationsgruppen bedeutet, dass die beiden Untergruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1},\Pi _{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi _{1},\Pi _{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/faf24582c5c412aa6eff24304263c42c1b1c4fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.629ex; height:2.509ex;" alt="{\displaystyle \Pi _{1},\Pi _{2}}" loading="lazy"> in der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy"> (im Sinne der <a href="#Operation_durch_Konjugation">Gruppenkonjugation</a>) zueinander konjugiert sind, nämlich durch eben jenes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \in \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi \in \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617bed41b08ae86d8f9f0738734aebd8be079a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.764ex; height:2.843ex;" alt="{\displaystyle \Phi \in \operatorname {Sym} (X)}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{2}=\Phi \circ \Pi _{1}\circ \Phi ^{-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi _{2}=\Phi \circ \Pi _{1}\circ \Phi ^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa528e1aed1f28d12871c0132d2bf16eaadd7e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.772ex; height:3.009ex;" alt="{\displaystyle \Pi _{2}=\Phi \circ \Pi _{1}\circ \Phi ^{-1}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Äquivalente_Permutationsdarstellungen">Äquivalente Permutationsdarstellungen</h4></div>

Zwei Permutationsdarstellungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{i}\colon G\rightarrow \operatorname {Sym} (X_{i})\;(i=1,2)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{i}\colon G\rightarrow \operatorname {Sym} (X_{i})\;(i=1,2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e509b18069e881655269d2ebc68c18b8f802a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.786ex; height:2.843ex;" alt="{\displaystyle \Theta _{i}\colon G\rightarrow \operatorname {Sym} (X_{i})\;(i=1,2)}" loading="lazy"> bzw. die durch sie induzierten Operationen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright _{i}\colon G\times X_{i}\to X_{i},\;g\triangleright _{i}x_{i}:=\Theta _{i}(g)(x_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright _{i}\colon G\times X_{i}\to X_{i},\;g\triangleright _{i}x_{i}:=\Theta _{i}(g)(x_{i})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b0ab1670f21b38a9d2e150833afde280265b526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.861ex; height:2.843ex;" alt="{\displaystyle \triangleright _{i}\colon G\times X_{i}\to X_{i},\;g\triangleright _{i}x_{i}:=\Theta _{i}(g)(x_{i})}" loading="lazy"> heißen äquivalent, wenn es eine Bijektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \colon X_{1}\to X_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi \colon X_{1}\to X_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72333cc1656f706fb3ddff74724e5dacbc0dd2c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.283ex; height:2.509ex;" alt="{\displaystyle \Phi \colon X_{1}\to X_{2}}" loading="lazy"> gibt, welche zugleich ein Homomorphismus (also ein Isomorphismus) zwischen den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f70b2694445a5901b24338a2e7a7e58f02a72a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"> ist, das heißt, welche sich in der folgenden Weise mit den Operationen „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright _{1},\triangleright _{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright _{1},\triangleright _{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5d4948ba8c4c9ba5231b96062d0a7df0134b16a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.467ex; height:2.176ex;" alt="{\displaystyle \triangleright _{1},\triangleright _{2}}" loading="lazy">“ auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f70b2694445a5901b24338a2e7a7e58f02a72a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"> verträgt: Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G,x_{1}\in X_{1},x_{2}\in X_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G,x_{1}\in X_{1},x_{2}\in X_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93d9e64dc61a59acd0543dbaec67a8417f596b73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.257ex; height:2.509ex;" alt="{\displaystyle g\in G,x_{1}\in X_{1},x_{2}\in X_{2}}" loading="lazy"> gilt

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi \left(g\triangleright _{1}x_{1}\right)=g\triangleright _{2}\Phi (x_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25a02846aafbc28523e0ac7313d57ebf3f086b7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.958ex; height:2.843ex;" alt="{\displaystyle \Phi \left(g\triangleright _{1}x_{1}\right)=g\triangleright _{2}\Phi (x_{1})}" loading="lazy"> oder äquivalent</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright _{1}x_{1}=\Phi ^{-1}\left(g\triangleright _{2}\Phi (x_{1})\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0513f131e101b61184953a1ba4e6240771e17ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.291ex; height:3.176ex;" alt="{\displaystyle g\triangleright _{1}x_{1}=\Phi ^{-1}\left(g\triangleright _{2}\Phi (x_{1})\right)}" loading="lazy">.</li></ul>

Ist dabei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=X_{2}=X}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1}=X_{2}=X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64b68d46539eefa70480bb415b45fe10de032b8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.134ex; height:2.509ex;" alt="{\displaystyle X_{1}=X_{2}=X}" loading="lazy">, so bedeutet die Äquivalenz zweier Permutationsdarstellungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{1},\Theta _{2}\colon G\to \operatorname {Sym} (X)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{1},\Theta _{2}\colon G\to \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53c283f3a05c099c38166aecd15555a0e27d5606.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.479ex; height:2.843ex;" alt="{\displaystyle \Theta _{1},\Theta _{2}\colon G\to \operatorname {Sym} (X)}" loading="lazy">, dass die beiden Untergruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{1}(G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{1}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f37068f9c2cc058001dab072cc947e12638f3ccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.498ex; height:2.843ex;" alt="{\displaystyle \Theta _{1}(G)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{2}(G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{2}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c0d0e94152c82d06ee0359025d3dc94e7d77b79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.498ex; height:2.843ex;" alt="{\displaystyle \Theta _{2}(G)}" loading="lazy"> in der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy"> (im Sinne der <a href="#Operation_durch_Konjugation">Gruppenkonjugation</a>) zueinander konjugiert sind, nämlich durch eben jenes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \in \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi \in \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617bed41b08ae86d8f9f0738734aebd8be079a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.764ex; height:2.843ex;" alt="{\displaystyle \Phi \in \operatorname {Sym} (X)}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall g\in G\colon \Theta _{1}(g)=\Phi ^{-1}\circ \Theta _{2}(g)\circ \Phi }">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall g\in G\colon \Theta _{1}(g)=\Phi ^{-1}\circ \Theta _{2}(g)\circ \Phi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07e82b14fe3ee23cd012d674f43eb9593eddc74d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.862ex; height:3.176ex;" alt="{\displaystyle \forall g\in G\colon \Theta _{1}(g)=\Phi ^{-1}\circ \Theta _{2}(g)\circ \Phi }" loading="lazy">.<a href="#cite_note-12">[8]</a>

<dl><dt>Beispiel</dt>

<dd>Bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon g\mapsto h*g*h^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon g\mapsto h*g*h^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6049ee575e16f54d15947188389403fd301cc25a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.8ex; height:3.176ex;" alt="{\displaystyle \varphi \colon g\mapsto h*g*h^{-1}}" loading="lazy"> einen <a href="Innerer_Automorphismus" class="mw-redirect" title="Innerer Automorphismus">inneren Automorphismus</a> auf einer Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, die vermöge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> operiert, so erhält man durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright ^{\varphi }x:=\varphi (g)\triangleright x=(h*g*h^{-1})\triangleright x=\left(\Theta _{\triangleright }(h)\circ \Theta _{\triangleright }(g)\circ \Theta _{\triangleright }(h)^{-1}\right)(x)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright ^{\varphi }x:=\varphi (g)\triangleright x=(h*g*h^{-1})\triangleright x=\left(\Theta _{\triangleright }(h)\circ \Theta _{\triangleright }(g)\circ \Theta _{\triangleright }(h)^{-1}\right)(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff9e9eec6478e44051e74598aca9a22c6e946366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:69.897ex; height:3.343ex;" alt="{\displaystyle g\triangleright ^{\varphi }x:=\varphi (g)\triangleright x=(h*g*h^{-1})\triangleright x=\left(\Theta _{\triangleright }(h)\circ \Theta _{\triangleright }(g)\circ \Theta _{\triangleright }(h)^{-1}\right)(x)}" loading="lazy"> eine äquivalente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Operation.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

Die <a href="%C3%84quivalenzklasse" class="mw-redirect" title="Äquivalenzklasse">Äquivalenzklassen</a> der <a href="#Bahn">oben</a> eingeführten Äquivalenzrelation „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sim }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }" loading="lazy">“ sind genau die Bahnen oder Konjugationsklassen (unter der Gruppenoperation). Daraus folgt die

<dl><dd>Bahnengleichung: Die <a href="M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Mächtigkeit</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ist gleich der Summe über die Länge aller Bahnen.</dd></dl>

Dabei gilt (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2f78101676239fc50bea86722dedf1512180e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle G_{x}}" loading="lazy"> als der <a href="#Stabilisator">Fixgruppe</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">) für jede Bahn (das heißt für jede Äquivalenzklasse) der

<dl><dd><a href="Bahnformel" title="Bahnformel">Bahnensatz</a>: Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> ist die (wohldefinierte!) Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon G/G_{x}\to G\triangleright x;\quad g*G_{x}\mapsto g\triangleright x,}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\colon G/G_{x}\to G\triangleright x;\quad g*G_{x}\mapsto g\triangleright x,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4abb357a574cb0286216f747b56aa5085bd1adcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.358ex; height:2.843ex;" alt="{\displaystyle i\colon G/G_{x}\to G\triangleright x;\quad g*G_{x}\mapsto g\triangleright x,}" loading="lazy"> eine <a href="Bijektivit%C3%A4t" class="mw-redirect" title="Bijektivität">Bijektion</a> von der Menge der Linksnebenklassen nach der Fixgruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in dessen Konjugationsklasse (Bahn).</dd></dl>

Für Rechtsoperationen sind stattdessen die Rechtsnebenklassen zu betrachten.

Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> endlich ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subset X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\subset X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdbc9a9f5d24b8a93789c81c1cb58b3305cc5734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle R\subset X}" loading="lazy"> ein Repräsentantensystem der Äquivalenzklassen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> bezeichne, dann lässt sich die obige Bahnengleichung folgendermaßen schreiben:

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \#X=\sum _{x\in R}G\triangleright x=\sum _{x\in R}(G:G_{x})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d327ec0dd89084664a796a49c1c61c3d7e90d4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.133ex; height:5.676ex;" alt="{\displaystyle \#X=\sum _{x\in R}G\triangleright x=\sum _{x\in R}(G:G_{x})}" loading="lazy"></dd></dl>

Aus der Bijektion des Bahnensatzes folgt für eine endliche Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> die <a href="Bahnformel" title="Bahnformel">Bahnformel</a>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G\triangleright x|\cdot |G_{x}|=|G|.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b3d69e8d06756422fa6ce9f56cbad732191ed23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.482ex; height:2.843ex;" alt="{\displaystyle |G\triangleright x|\cdot |G_{x}|=|G|.}" loading="lazy"></dd></dl>

Insbesondere ist die Länge jeder Bahn ein Teiler der Ordnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Permutationsgruppen">Permutationsgruppen</h3></div>

Die beiden oben bereits beispielhaft genannten Gruppen

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{1}:=\{e,(12)(34),(13)(24),(14)(23)\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49c40c6cce5f5a5f5d30b9cde6ee16be40368b6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.47ex; height:2.843ex;" alt="{\displaystyle V_{1}:=\{e,(12)(34),(13)(24),(14)(23)\}}" loading="lazy"> und</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{2}:=\{e,(12),(34),(12)(34)\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/967f3bb99606f219632f17e55f6535837dd33b02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.202ex; height:2.843ex;" alt="{\displaystyle V_{2}:=\{e,(12),(34),(12)(34)\}}" loading="lazy"></li></ul>

sind als Gruppen isomorph zu <a href="Kleinsche_Vierergruppe" title="Kleinsche Vierergruppe">(Kleinschen) Vierergruppe</a>.

Ihre Operation auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\{1,2,3,4\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:=\{1,2,3,4\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c88120d298d5185fcdc5b9b8edaa09f8cb79aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.802ex; height:2.843ex;" alt="{\displaystyle X:=\{1,2,3,4\}}" loading="lazy"> sind nicht äquivalent:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adfdbc929f16cb00bb43289c223651b41f7b9f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}" loading="lazy"> operiert (scharf einfach) transitiv, da die Ziffer 1 in jede andere (von genau einem Gruppenelement) überführt werden kann. Darüber hinaus operiert sie imprimitiv, wie die möglichen Partitionen in Imprimitivitätsgebiete (nichttriviale Kongruenzklassen) zeigen:

<ul><li>Entweder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\{1,2\}\uplus \{3,4\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:=\{1,2\}\uplus \{3,4\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7360731acf59db7a442972c413ca62b16c8bfdd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.675ex; height:2.843ex;" alt="{\displaystyle X:=\{1,2\}\uplus \{3,4\}}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:=\{1,3\}\uplus \{2,4\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f44f3137e7215399d676766853caa6a1df6505f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.675ex; height:2.843ex;" alt="{\displaystyle X:=\{1,3\}\uplus \{2,4\}}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:=\{1,4\}\uplus \{2,3\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bed36c2c1e5226e033fa4ab3c6ecb14b6416d0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.675ex; height:2.843ex;" alt="{\displaystyle X:=\{1,4\}\uplus \{2,3\}}" loading="lazy">.</li></ul></li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceaa689a894f5020a7b46177d201cbce2d41122b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{2}}" loading="lazy"> hingegen operiert nicht transitiv. Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> zerfällt in die Bahnen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,2\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/111aef9df2c1a71ed7820a5426ef3e8c7b781564.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{1,2\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{3,4\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{3,4\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ce803a141c2412f297a65b77a43d94ce3d17b27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{3,4\}}" loading="lazy">.</li></ul>

Beachte hierbei, dass die ursprünglich treue Operation der Kleinschen Vierergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adfdbc929f16cb00bb43289c223651b41f7b9f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}" loading="lazy"> bei der Reduktion auf die beiden Imprimitivitätsgebiete aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/\!\!\sim }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X/\!\!\sim }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3632d6132fb4dee8c1b03f277d980ef1d73b26f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle X/\!\!\sim }" loading="lazy"> ihre Treue verliert.

Äquivalente Permutationsgruppendarstellungen setzen gleiche Transformationsgruppen voraus. Isomorphe Permutationsgruppen können jedoch inäquivalent sein. Im vorliegenden Falle gibt es keine bijektive Selbstabbildung auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, die zwischen beiden Operationen vermittelt.

Beispielsweise verfügt die alternierende Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e213bbb69691c65e1391fe16cd79a0029471446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{5}}" loading="lazy"> fünften Grades über transitive primitive Darstellungen fünften, sechsten und zehnten Grades, die natürlich nicht äquivalent sind. Weitere transitive primitive Darstellungen anderer Grade gibt es allerdings nicht.<a href="#cite_note-13">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Darstellung_durch_Permutationsmatrizen">Darstellung durch Permutationsmatrizen</h3></div>

Operiert die Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf einer Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> eines endlichdimensionalen <a href="Vektorraum" title="Vektorraum">Vektorraumes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> über dem <a href="K%C3%B6rper" class="mw-disambig" title="Körper">Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und setzt man die Permutationen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy"> linear auf den Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> fort, so erhält man eine <a href="Darstellungstheorie" title="Darstellungstheorie">lineare Darstellung</a> der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> durch Automorphismen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, die (bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">) durch <a href="Permutationsmatrix" title="Permutationsmatrix">Permutationsmatrizen</a> dargestellt werden und natürlich bijektiv sind: Vgl. auch <a href="Permutationsgruppe#Permutationsdarstellung" title="Permutationsgruppe">lineare Permutationsdarstellung</a>.

<div class="mw-heading mw-heading3"><h3 id="Operation_einer_Gruppe_auf_sich_selbst">Operation einer Gruppe auf sich selbst</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Operation_durch_Multiplikation">Operation durch Multiplikation</h4></div>

Das einfachste Beispiel einer Operation ist die Operation einer Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> auf sich selbst: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy"> ist stets eine Operation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g*h)*x=g*(h*x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g*h)*x=g*(h*x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffbd089856daa88c41a70df7b8cb9e29c99f28a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.065ex; height:2.843ex;" alt="{\displaystyle (g*h)*x=g*(h*x)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e*g=g.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e*g=g.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0a912ddf79b30a1c6399bde52d3991f750776.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.256ex; height:2.009ex;" alt="{\displaystyle e*g=g.}" loading="lazy">

Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta \colon G\to \operatorname {Sym} (G)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta \colon G\to \operatorname {Sym} (G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6b4a3974039e94c2a1d219ea703be7c68b9c208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.375ex; height:2.843ex;" alt="{\displaystyle \Theta \colon G\to \operatorname {Sym} (G)}" loading="lazy"> ordnet jedem Gruppenelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> die Linkstranslation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{g*}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta _{g*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff60b2e9bbc09ff76306c54f9a4791fb459bbab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.217ex; height:2.843ex;" alt="{\displaystyle \vartheta _{g*}}" loading="lazy"> mit diesem zu. Weil die Operation treu ist, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Theta }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\Theta }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9b5471f462729378c031ac61b773398c6772dd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle {\Theta }}" loading="lazy"> ein <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektiver</a> Gruppenhomomorphismus, man erhält hieraus den

<dl><dd><a href="Satz_von_Cayley" title="Satz von Cayley">Satz von Cayley</a>: Jede <a href="Endliche_Gruppe" title="Endliche Gruppe">endliche Gruppe</a> der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> ist isomorph zu einer Untergruppe der symmetrischen Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} _{n\!}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} _{n\!}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e2db5b31b0b20c502ac2a0b8b5076479be152b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.934ex; height:2.676ex;" alt="{\displaystyle \operatorname {Sym} _{n\!}.}" loading="lazy"></dd></dl>

Mit dieser Untergruppe der symmetrischen Gruppe ist also Darstellung der Gruppe als Permutationsgruppe (als Gruppe von Permutationen auf einer symmetrischen Gruppe) gefunden. Im obigen Falle hat die symmetrische Gruppe den Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G:1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f97adaa7944d8f59526ca587d5cb119ab98756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.736ex; height:2.843ex;" alt="{\displaystyle (G:1)}" loading="lazy">. Der nächste Abschnitt zeigt, dass sich diese Anzahl unter günstigen Umständen verringern lässt.

Analoges gilt auch für die Rechtstranslation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{*g}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta _{*g}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07fa4416fed5ee5f2825a4f4186636bae8d4ca5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.864ex; height:2.843ex;" alt="{\displaystyle \vartheta _{*g}.}" loading="lazy">

Betrachtet man eine Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> dann operiert auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy"> Die Bahn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H*g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H*g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23bf1ea9b0dbda3d13f28b55097d5d9b6d6367a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.374ex; height:2.509ex;" alt="{\displaystyle H*g}" loading="lazy"> eines Elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> heißt dann auch Rechtsnebenklasse und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e205907f4c401e0de8a75ea1b3dd76b14e5bdc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.374ex; height:2.509ex;" alt="{\displaystyle g*H}" loading="lazy"> Linksnebenklasse von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23a3f421f58ef3bc6f9ec70e883e1496ff871e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g.}" loading="lazy"> Man beachte, dass im Allgemeinen nicht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*H=H*g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*H=H*g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b1699dd70290d9e9d173c92f8e55f037d4088e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.847ex; height:2.509ex;" alt="{\displaystyle g*H=H*g}" loading="lazy"> sein muss. Die Mächtigkeit der Menge aller Rechtsnebenklassen bezeichnet man mit

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:H:=|H\backslash G|.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36263a8f8a8b02e10a8b06b098b56cb9def72278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.566ex; height:2.843ex;" alt="{\displaystyle G:H:=|H\backslash G|.}" loading="lazy"></dd></dl>

Da in einer Gruppe jede Rechtstranslation eine Bijektion ist, gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H*g|=|H|}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |H*g|=|H|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fd878d205ce6ccb01433a29dfe52673c4fa095b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.124ex; height:2.843ex;" alt="{\displaystyle |H*g|=|H|}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0881a8551fe99fd948d4f9fefe10ee7125619019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.43ex; height:2.509ex;" alt="{\displaystyle g\in G.}" loading="lazy"> Daraus folgt mit der Bahnengleichung der

<dl><dd><a href="Satz_von_Lagrange" title="Satz von Lagrange">Satz von Euler-Lagrange</a>: Für jede Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> einer endlichen Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> gilt:

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G|=(G:H)\cdot |H|.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89a54f103c15c943550f13dfbf941c212a8564ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.539ex; height:2.843ex;" alt="{\displaystyle |G|=(G:H)\cdot |H|.}" loading="lazy"></dd></dl></dd>

<dd>Insbesondere ist die Ordnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> ein Teiler der Ordnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy"></dd></dl>

Man kann zeigen, dass es genauso viele Linksnebenklassen wie Rechtsnebenklassen gibt, dass also

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:H=|H\backslash G|=|G/H|.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceda78efd3513866cb85a547d5066123a7581507.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.364ex; height:2.843ex;" alt="{\displaystyle G:H=|H\backslash G|=|G/H|.}" loading="lazy"></dd></dl>

Eine Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> heißt <a href="Normalteiler" title="Normalteiler">Normalteiler</a>, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*H=H*g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*H=H*g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b1699dd70290d9e9d173c92f8e55f037d4088e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.847ex; height:2.509ex;" alt="{\displaystyle g*H=H*g}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> gilt. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> ein Normalteiler von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> dann wird durch

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g_{1}*H)\circledast (g_{2}*H):=(g_{1}*g_{2})*H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02cc0eab9b4fd9436593a0bfcdf44cc9850c85df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.636ex; height:2.843ex;" alt="{\displaystyle (g_{1}*H)\circledast (g_{2}*H):=(g_{1}*g_{2})*H}" loading="lazy"></dd></dl>

eine Verknüpfung auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> definiert, mit der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G/H,\circledast )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G/H,\circledast )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c17695382ec0a94b2a925bf4e80ba04e7c1be9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.704ex; height:2.843ex;" alt="{\displaystyle (G/H,\circledast )}" loading="lazy"> eine Gruppe ist, man nennt sie die <a href="Faktorgruppe" title="Faktorgruppe">Faktorgruppe</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8933ae7244305ae7824aa18e077d1cf946e2ee9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.71ex; height:2.176ex;" alt="{\displaystyle H.}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Operation_auf_den_Nebenklassen_durch_Multiplikation">Operation auf den Nebenklassen durch Multiplikation</h4></div>

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> eine Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> bezeichne die Menge der Linksnebenklassen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*H\;(g\in G)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*H\;(g\in G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53896f854df1592e40e54520f22a654c5b682662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.612ex; height:2.843ex;" alt="{\displaystyle g*H\;(g\in G)}" loading="lazy">. Dann operiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> in naheliegender Weise auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \triangleright (g*H):=(\gamma *g)*H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \triangleright (g*H):=(\gamma *g)*H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5004c44a22ca2889cc93d09802002f9d123b6f18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.027ex; height:2.843ex;" alt="{\displaystyle \gamma \triangleright (g*H):=(\gamma *g)*H}" loading="lazy">. Diese Operation ist transitiv. Sie liefert also (gemäß obiger Feststellung, dass jede Gruppenoperation als Operation einer Permutationsgruppe auffassen lässt) einen Gruppenhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{H}\colon G\to \operatorname {Sym} (G/H),\;\gamma \mapsto [g*H\mapsto (\gamma *g)*H)]}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{H}\colon G\to \operatorname {Sym} (G/H),\;\gamma \mapsto [g*H\mapsto (\gamma *g)*H)]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5859b707e9cd8198ff442da3f0b43676e2c85f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.675ex; height:2.843ex;" alt="{\displaystyle \Theta _{H}\colon G\to \operatorname {Sym} (G/H),\;\gamma \mapsto [g*H\mapsto (\gamma *g)*H)]}" loading="lazy">. Für den Kern des Homomorphismus gilt:

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ker \Theta _{H}=\{\gamma \in G\mid \forall g\in G:\gamma *g\in g*H\}=\bigcap _{g\in G}g*H*g^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddd4b11618d6df81af277c19106531bc3878331d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:59.832ex; height:5.843ex;" alt="{\displaystyle \ker \Theta _{H}=\{\gamma \in G\mid \forall g\in G:\gamma *g\in g*H\}=\bigcap _{g\in G}g*H*g^{-1}}" loading="lazy">.</dd></dl>

Im Falle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker \Theta _{H}=\{e_{G}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ker \Theta _{H}=\{e_{G}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c35a45769b65f8fef1bdb9893f0805256f53b01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.089ex; height:2.843ex;" alt="{\displaystyle \ker \Theta _{H}=\{e_{G}\}}" loading="lazy"> liefert die Operation auf den Linksnebenklassen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> bereits eine Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> als Permutationsgruppe, nämlich als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{H}(G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{H}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9332469f4b4b93dca51f928ebfa1b30f878e8660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.136ex; height:2.843ex;" alt="{\displaystyle \Theta _{H}(G)}" loading="lazy">.<a href="#cite_note-14">[10]</a>

Im Trivialfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\{e_{G}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=\{e_{G}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91cd6b149b92545dce21dc86d6e3ad978db3ee80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.094ex; height:2.843ex;" alt="{\displaystyle H=\{e_{G}\}}" loading="lazy">, der oben im Abschnitt „<a href="#Operation_durch_Multiplikation">Operation durch Multiplikation</a>“ betrachtet wurde, besitzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{e}=\Theta _{\triangleright }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{e}=\Theta _{\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c32629deb1403a926fca2ee4204b6fb207b106e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.767ex; height:2.509ex;" alt="{\displaystyle \Theta _{e}=\Theta _{\triangleright }}" loading="lazy"> trivialen Kern: Die treue Operation liefert eine Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> als Permutationsgruppe vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G:1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f97adaa7944d8f59526ca587d5cb119ab98756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.736ex; height:2.843ex;" alt="{\displaystyle (G:1)}" loading="lazy">, wie oben im Satz von Cayley erkannt wurde.

Doch eine „größere“ Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> könnte günstigenfalls (nämlich bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{H}=\{e\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{H}=\{e\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4ef57b032060eccbdf707657cc17d8d2f3e92a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.006ex; height:2.843ex;" alt="{\displaystyle \Theta _{H}=\{e\}}" loading="lazy">) den Grad der Permutationsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{H}(G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{H}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9332469f4b4b93dca51f928ebfa1b30f878e8660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.136ex; height:2.843ex;" alt="{\displaystyle \Theta _{H}(G)}" loading="lazy"> – das ist die Anzahl der zu permutierenden Objekte – von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G:1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f97adaa7944d8f59526ca587d5cb119ab98756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.736ex; height:2.843ex;" alt="{\displaystyle (G:1)}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G:H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/902f5b36c3c66fa67326ed607480bc4ef8ae6bff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.637ex; height:2.843ex;" alt="{\displaystyle (G:H)}" loading="lazy"> reduzieren und trotzdem noch eine Darstellung der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> als Permutationsgruppe liefern.

Tatsächlich empfiehlt sich hierfür die Fixgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=G_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=G_{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a9795e7129821f5769e7268ddee6f804ae29c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.161ex; height:2.509ex;" alt="{\displaystyle H=G_{x}}" loading="lazy"> eines Elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d7e0b51bd905f35d11790939139d18014f8b017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.997ex; height:2.176ex;" alt="{\displaystyle x\in G}" loading="lazy">: Der Transitivität wegen sind alle Gruppenelemente zueinander konjugiert, mithin sind ihre Fixgruppen die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> konjugierten Untergruppen; der Treue wegen ist der Durchschnitt all dieser Fixgruppen trivial, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{H}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{H}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ccb89fc3605e204ff7bd268e45eb1e836938fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.5ex; height:2.509ex;" alt="{\displaystyle \Theta _{H}}" loading="lazy"> eine Permutationsgruppendarstellung vom Grade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G:H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/902f5b36c3c66fa67326ed607480bc4ef8ae6bff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.637ex; height:2.843ex;" alt="{\displaystyle (G:H)}" loading="lazy"> ist.

Für konjugierte Untergruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H'=h^{-1}*H*h}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H'=h^{-1}*H*h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88e7b8a2fd39ef5d61d7f0d0f270f15ca2aeccae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.35ex; height:2.676ex;" alt="{\displaystyle H'=h^{-1}*H*h}" loading="lazy"> (mit einem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c938e8bf0bc7fa38baf00c88fa8aa6acf633f823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle h\in G}" loading="lazy">) erhält man freilich zwei verschiedene Permutationsgruppendarstellungen, da die Mengen der Linksnebenklassen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H'}">

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/H'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/683ff9b11386853a65b807ba35c6ac6abeac18bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.777ex; height:3.009ex;" alt="{\displaystyle G/H'}" loading="lazy"> unterschiedlich sein können. Doch dürfen sie in jedem Falle als äquivalent gelten, denn der innere Automorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{h}\colon g\mapsto h^{-1}*g*h}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iota _{h}\colon g\mapsto h^{-1}*g*h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23940d05e9fd3735bb635fa6e35ecd1f3fe1321a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.283ex; height:3.009ex;" alt="{\displaystyle \iota _{h}\colon g\mapsto h^{-1}*g*h}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{h}(H)=H'}">

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iota _{h}(H)=H'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b488a95289bafa3f5204e56de30d05dfab66c75d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.762ex; height:3.009ex;" alt="{\displaystyle \iota _{h}(H)=H'}" loading="lazy">) induziert die folgende Bijektion zwischen den beiden Mengen:

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mrow>

</mrow>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</msup>

</mrow>

</munder>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\iota _{h}}}\colon G/H\to G/H',\;g*H\longmapsto h^{-1}*g*H*h=\underbrace {(h^{-1}*g*h)} _{\iota _{h}(g)}*\underbrace {(h^{-1}*H*h)} _{=\iota _{h}(H)=H'}\in G/H'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ffd5bfe9b06ad9f8a1633befa956bf79ebcb926.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:84.029ex; height:7.176ex;" alt="{\displaystyle {\overline {\iota _{h}}}\colon G/H\to G/H',\;g*H\longmapsto h^{-1}*g*H*h=\underbrace {(h^{-1}*g*h)} _{\iota _{h}(g)}*\underbrace {(h^{-1}*H*h)} _{=\iota _{h}(H)=H'}\in G/H'}" loading="lazy"></dd></dl>

Diese Bijektion vermittelt zwischen beiden Permutationsdarstellungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{H}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{H}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ccb89fc3605e204ff7bd268e45eb1e836938fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.5ex; height:2.509ex;" alt="{\displaystyle \Theta _{H}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{H'}}">

<msub>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{H'}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18227ac601227d1a7809e2c2fc5f487d10572094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.059ex; height:2.676ex;" alt="{\displaystyle \Theta _{H'}}" loading="lazy"> und zeigt, dass sie äquivalent sind, also im Wesentlichen gleich, bis auf das Aussehen der zu permutierenden Objekte und einen inneren Automorphismus der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

Für Rechtsnebenklassen bei Rechtsoperation gelten selbstverständlich analoge Überlegungen.<a href="#cite_note-15">[11]</a>

<div class="mw-heading mw-heading4"><h4 id="Operation_durch_Konjugation">Operation durch Konjugation</h4></div>

Eine Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> operiert auf sich selbst durch die <a href="Automorphismus#Innere_Automorphismen" title="Automorphismus">inneren Automorphismen</a>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{g}\colon G\to G,\;h\mapsto g*h*g^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e02ae7bc5d97199c40880896b756917199c878fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.771ex; height:3.343ex;" alt="{\displaystyle \varphi _{g}\colon G\to G,\;h\mapsto g*h*g^{-1}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy">.</dd></dl>

Man erhält den Gruppenhomomorphismus

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon G\to \operatorname {Aut} (G),\quad g\mapsto \varphi _{g}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/114afa1949d4ffdf08e76250c4388acc93cf9401.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.199ex; height:3.009ex;" alt="{\displaystyle \varphi \colon G\to \operatorname {Aut} (G),\quad g\mapsto \varphi _{g}}" loading="lazy">.</dd></dl>

Die Menge aller inneren Automorphismen wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Inn} (G):=\varphi (G)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Inn} (G):=\varphi (G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa1dc08da9452fba16234f109a1a98844eac1546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.962ex; height:2.843ex;" alt="{\displaystyle \operatorname {Inn} (G):=\varphi (G)}" loading="lazy"> bezeichnet.

Die zugrunde liegende Gruppenoperation<a href="#cite_note-16">[Anm 5]</a> ist also: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\triangleright h:=\varphi _{g}(h)=g*h*g^{-1}.}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright h:=\varphi _{g}(h)=g*h*g^{-1}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e563eb523d72651b3f1e0296725f8a5b6ab9c98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.126ex; height:3.343ex;" alt="{\displaystyle g\triangleright h:=\varphi _{g}(h)=g*h*g^{-1}.}" loading="lazy">

Tatsächlich wurde diese Operation einer Gruppe auf sich selbst in der klassischen Gruppentheorie als die <a href="Konjugation_(Gruppentheorie)" title="Konjugation (Gruppentheorie)">Konjugation</a> (von Gruppenelementen) bezeichnet. Daher spricht man in der <a href="Gruppentheorie" title="Gruppentheorie">Gruppentheorie</a> üblicherweise von den <a href="Konjugationsklasse" class="mw-redirect" title="Konjugationsklasse">Konjugationsklassen</a> (anstelle von Bahnen) und des Weiteren vom <a href="Zentralisator" title="Zentralisator">Zentralisator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{G}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{G}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95e703e1c5b5bbbae49e929113903dd5b68f246a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle Z_{G}(x)}" loading="lazy"> anstelle des Stabilisators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2f78101676239fc50bea86722dedf1512180e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle G_{x}}" loading="lazy">. Die <a href="Bahnformel" title="Bahnformel">Bahnformel</a> lautet

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \#\,(G\triangleright x)=(G:Z_{G}(x))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7313c045b7f3663cfe65ac48ea77fcd524719bcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.405ex; height:2.843ex;" alt="{\displaystyle \#\,(G\triangleright x)=(G:Z_{G}(x))}" loading="lazy">, falls beide Seiten endlich sind,</dd></dl>

und die Bahnengleichung heißt in diesem Kontext üblicherweise <a href="Klassengleichung" class="mw-redirect" title="Klassengleichung">Klassengleichung</a>, setzt eine endliche Operatorgruppe voraus und lautet:

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:1)=\sum _{x\in R}G\triangleright x=\sum _{x\in R}(G:Z_{G}(x))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d3b4bc4da1d1dd4392fc8bea7505a679490fc8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.204ex; height:5.676ex;" alt="{\displaystyle (G:1)=\sum _{x\in R}G\triangleright x=\sum _{x\in R}(G:Z_{G}(x))}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subset G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\subset G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa1d1ea4d8024fab2b0bb0ba9c46dea675e4b388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.689ex; height:2.176ex;" alt="{\displaystyle R\subset G}" loading="lazy"> ein Repräsentantensysteme der Konjugationsklassen bezeichne.</dd></dl>

Eine elegante Anwendung der <a href="Klassengleichung" class="mw-redirect" title="Klassengleichung">Klassengleichung</a> lieferte <a href="Ernst_Witt" title="Ernst Witt">Ernst Witt</a> mit <a href="Satz_von_Wedderburn#Ernst_Witt_(1931)" title="Satz von Wedderburn">seinem kurzen Beweis</a> (1931) des (kleinen) <a href="Satz_von_Wedderburn" title="Satz von Wedderburn">Satzes von Wedderburn (1905)</a>: „Endliche Schiefkörper sind kommutativ.“

<div class="mw-heading mw-heading3"><h3 id="Automorphismengruppe_eines_Körpers_und_einer_Galois-Erweiterung">Automorphismengruppe eines Körpers und einer Galois-Erweiterung</h3></div>

Die Gruppe der Automorphismen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{0}:=\operatorname {Aut} (L)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{0}:=\operatorname {Aut} (L)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48350858af4b6bd6fdcd690f511a4f501e340248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.959ex; height:2.843ex;" alt="{\displaystyle G_{0}:=\operatorname {Aut} (L)}" loading="lazy"> eines Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> operiert in offensichtlicher Weise auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\;(=X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L\;(=X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fee6ce45e4c37830fff3e71decc3860ed5248a66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.471ex; height:2.843ex;" alt="{\displaystyle L\;(=X)}" loading="lazy">, denn damit ist ja der erforderliche Gruppenhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta \colon G_{0}\hookrightarrow \operatorname {Sym} (L)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta \colon G_{0}\hookrightarrow \operatorname {Sym} (L)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55625299f4204a122c2128dc7480dcef0565d8de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.478ex; height:2.843ex;" alt="{\displaystyle \Theta \colon G_{0}\hookrightarrow \operatorname {Sym} (L)}" loading="lazy"> unmittelbar gegeben. Die Operation ist durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \triangleright y=\sigma (y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \triangleright y=\sigma (y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec2ad7ef00922b18f3edf436c7a1a071d8ccaeec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.073ex; height:2.843ex;" alt="{\displaystyle \sigma \triangleright y=\sigma (y)}" loading="lazy"> geben, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \in G_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \in G_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a73489a6e26ff7b4d7c9fcd274b0e7985ec379a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.051ex; height:2.509ex;" alt="{\displaystyle \sigma \in G_{0}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc54f30a0ec14db7f7f416a6f8f9ed651e6a676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.579ex; height:2.509ex;" alt="{\displaystyle y\in L}" loading="lazy">.

Es sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\subset G_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\subset G_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58a09736cb3b76778e0a0428a7d10c230451fea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.806ex; height:2.509ex;" alt="{\displaystyle G\subset G_{0}}" loading="lazy"> eine Untergruppe (bspw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=G_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=G_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6044301da7aa1628be72c6266e33a7ed63fc3f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.806ex; height:2.509ex;" alt="{\displaystyle G=G_{0}}" loading="lazy">) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K:=L^{G}:=\{y\in L\mid \forall \sigma \in G:\sigma (y)=y\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K:=L^{G}:=\{y\in L\mid \forall \sigma \in G:\sigma (y)=y\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a8d0a4a1542dea1f1654c7792a921580ac2ff91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.28ex; height:3.176ex;" alt="{\displaystyle K:=L^{G}:=\{y\in L\mid \forall \sigma \in G:\sigma (y)=y\}}" loading="lazy"> der Fixkörper in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> unter der Operation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. Wenn die Erweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> zudem algebraisch ist, dann ist sie eine <a href="Galois-Erweiterung" class="mw-redirect" title="Galois-Erweiterung">Galois-Erweiterung</a> mit der Galois-Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\operatorname {Aut} (L/K)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=\operatorname {Aut} (L/K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/423b7d93e96235c423a236fd18f8d538cfd51603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.486ex; height:2.843ex;" alt="{\displaystyle G=\operatorname {Aut} (L/K)}" loading="lazy">, und diese Erweiterung ist endlich, sobald <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [L:K]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [L:K]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aece5b0ecef53af76d4af496d68ed1c41532bb79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.88ex; height:2.843ex;" alt="{\displaystyle [L:K]}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G:1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G:1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f97adaa7944d8f59526ca587d5cb119ab98756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.736ex; height:2.843ex;" alt="{\displaystyle (G:1)}" loading="lazy"> endlich sind. In diesem Falle gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [L:K]=(G:1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [L:K]=(G:1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac31f23e3c27b398c543b54ba3eac98bdff87886.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.714ex; height:2.843ex;" alt="{\displaystyle [L:K]=(G:1)}" loading="lazy">.

Die Galois-Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\operatorname {Aut} (L/K)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=\operatorname {Aut} (L/K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/423b7d93e96235c423a236fd18f8d538cfd51603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.486ex; height:2.843ex;" alt="{\displaystyle G=\operatorname {Aut} (L/K)}" loading="lazy"> einer Galois-Erweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> operiert, wie oben gezeigt, auf dem Erweiterungskörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. Dabei sind die Elemente, die unter der Operation der Galoios-Gruppe konjugiert sind, genau diejenigen, die man kurz als über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> konjugiert bezeichnet: Sie sind dadurch gekennzeichnet, dass sie dasselbe <a href="Minimalpolynom_(K%C3%B6rpertheorie)" title="Minimalpolynom (Körpertheorie)">Minimalpolynom</a> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> besitzen. Dessen <a href="Nullstellenmenge" title="Nullstellenmenge">Nullstellenmenge</a> ist also eine Bahn unter der Operation der Galois-Gruppe und stellt eine Konjugationsklasse über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> bzw. unter der Operation der Galois-Gruppe dar. Gerade weil es eine Galois-Erweiterung ist, liegen nämlich alle seine Nullstellen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> (Eigenschaft der Normalität).<a href="#cite_note-17">[Anm 6]</a><a href="#cite_note-18">[12]</a>

Ist umgekehrt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\{y_{1},\dots y_{n}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=\{y_{1},\dots y_{n}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d59fff17d45894851cffb2e310570ce02cbbcb32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.892ex; height:2.843ex;" alt="{\displaystyle Y=\{y_{1},\dots y_{n}\}}" loading="lazy"> eine endliche, unter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> stabile Teilmenge von Körperelementen aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">, die also unter der Operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> permutiert werden, so liegen die Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"> des Polynoms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x):=(X-y_{1})\cdots (X-y_{n})=:X^{n}+a_{1}X^{n-1}+\dots a_{n-1}X+a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x):=(X-y_{1})\cdots (X-y_{n})=:X^{n}+a_{1}X^{n-1}+\dots a_{n-1}X+a_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e0604160d10eb6f9e8083d6ff7b29b167394256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.64ex; height:3.176ex;" alt="{\displaystyle f(x):=(X-y_{1})\cdots (X-y_{n})=:X^{n}+a_{1}X^{n-1}+\dots a_{n-1}X+a_{n}}" loading="lazy"> notwendig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, und es ist genau dann über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> irreduzibel, wenn sämtliche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> untereinander konjugiert sind, das heißt, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> transitiv auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> operiert. Andernfalls besteht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> aus mehr als nur einer Bahn (Konjugationsklasse), zerfällt also in eine Partition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\biguplus _{j=1}^{N}Y_{j}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=\biguplus _{j=1}^{N}Y_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50d654eef6a6e1257f435db9947a4d8a50a11b55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:10.297ex; height:7.676ex;" alt="{\displaystyle Y=\biguplus _{j=1}^{N}Y_{j}}" loading="lazy"> disjunkter Bahnen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{j}\;(j=1,\dots ,N\geq 2)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{j}\;(j=1,\dots ,N\geq 2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03a4cca9730300bf275e18ca4ffc6648142ffc44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.436ex; height:3.009ex;" alt="{\displaystyle Y_{j}\;(j=1,\dots ,N\geq 2)}" loading="lazy"> und das Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7425278aab7b6ceb8ddb54fa5ce71d2cf52d28f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.048ex; height:2.843ex;" alt="{\displaystyle p(X)}" loading="lazy"> zerfällt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> irreduziblen Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{j}(X)=\prod _{y\in Y_{j}}(X-y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{j}(X)=\prod _{y\in Y_{j}}(X-y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c45c80ae67d35b24f8aa62d95994cd666b021d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-left: -0.089ex; width:20.424ex; height:6.176ex;" alt="{\displaystyle p_{j}(X)=\prod _{y\in Y_{j}}(X-y)}" loading="lazy">.

All dies gilt insbesondere für ein – nach dem <a href="Satz_vom_primitiven_Element" title="Satz vom primitiven Element">Satz vom primitiven Element</a> (<a href="Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a>, 1829) existierendes – erzeugendes Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}\in L}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{1}\in L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e707d8e3b98f254e518398adcf9eec2e67bddc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.617ex; height:2.509ex;" alt="{\displaystyle y_{1}\in L}" loading="lazy"> einer endlichen Galois-Erweiterung mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=K[y_{1}]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=K[y_{1}]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/749bbe9a1c6815c24b1bc1876377beda1f2a569c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.234ex; height:2.843ex;" alt="{\displaystyle L=K[y_{1}]}" loading="lazy">: Seine sämtlichen Konjugierten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1},\dots ,y_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{1},\dots ,y_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd782cc196312fc3cdda55540db15b6e741818f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.729ex; height:2.009ex;" alt="{\displaystyle y_{1},\dots ,y_{n}}" loading="lazy"> (wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=[L:K]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=[L:K]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96b81d44ad43163032e8c59bf674878b85d0e19b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.373ex; height:2.843ex;" alt="{\displaystyle n=[L:K]}" loading="lazy">) sind genau die Nullstellen desselben Minimalpolynoms über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, welches den Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> besitzt (und das in historischer Sprechweise eine <a href="Galois-Resolvente" class="mw-redirect" title="Galois-Resolvente">Galois-Resolvente</a> ist), und bilden die Bahn unter der Operation der Galois-Gruppe: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\triangleright y_{i}=\{y_{1},\dots ,y_{n}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\triangleright y_{i}=\{y_{1},\dots ,y_{n}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d564812d5470b09c08eee9cc13b5d9b5082e781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.113ex; height:2.843ex;" alt="{\displaystyle G\triangleright y_{i}=\{y_{1},\dots ,y_{n}\}}" loading="lazy">. Ein Automorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae2ea6a3648b2b75cc8103c569a9630c194dcce1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.997ex; height:2.176ex;" alt="{\displaystyle \sigma \in G}" loading="lazy"> ist daher bereits durch sein Bild <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (y_{1})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (y_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae2f283100ce757324d762c8c1879b0e75ca2d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.332ex; height:2.843ex;" alt="{\displaystyle \sigma (y_{1})}" loading="lazy"> festgelegt. Dies heißt wiederum, dass die Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{1},\dots ,y_{n}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{y_{1},\dots ,y_{n}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/346cae1bbb2cb806d142ca928b00ef2c5fce1cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.054ex; height:2.843ex;" alt="{\displaystyle \{y_{1},\dots ,y_{n}\}}" loading="lazy"> (also auf der Nullstellenmenge der Galois-Resolvente) scharf einfach transitiv operiert. Die Operation ist also frei, mithin erst recht treu.<a href="#cite_note-19">[Anm 7]</a>

Für einen Zwischenkörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\;(L\supset Z\supset K)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\;(L\supset Z\supset K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a247d107e9bc50d4e706b6146859c166b2a98e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.661ex; height:2.843ex;" alt="{\displaystyle Z\;(L\supset Z\supset K)}" loading="lazy"> definiert man (analog zu obiger Definition):

</mrow>

</mrow>

<mtext> ist Fixpunktmenge unter </mtext>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(L/Z):=\operatorname {Aut} (L/Z)=\{\sigma \in G\mid Z{\text{ ist Fixpunktmenge unter }}\sigma \}=\bigcap _{z\in Z}G_{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f51f05fd3ae23990cd4bd714d55d4f117a0d5a4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:73.799ex; height:5.676ex;" alt="{\displaystyle G(L/Z):=\operatorname {Aut} (L/Z)=\{\sigma \in G\mid Z{\text{ ist Fixpunktmenge unter }}\sigma \}=\bigcap _{z\in Z}G_{z}}" loading="lazy">.</dd></dl>

Für die Galoiserweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> und den Verband <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {Z}}(L/K)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Z}}(L/K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad92a4d98a38459570fde7930be24821f9526b7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.02ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {Z}}(L/K)}" loading="lazy"> ihrer Zwischenkörper einerseits sowie für die Galoisgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=:G(L/K)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=:G(L/K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f1273b70beb88c4e7dfb6dfb6193781b2cbe0a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.019ex; height:2.843ex;" alt="{\displaystyle G=:G(L/K)}" loading="lazy"> und den Verband <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}(G)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {U}}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50f29fa93aa8b883eb89017ae8581e703de910ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.057ex; width:5.239ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {U}}(G)}" loading="lazy"> ihrer Untergruppen andererseits lässt sich unter Verwendung der oben genannten Begrifflichkeiten der <a href="Galoistheorie#Hauptsatz_der_Galoistheorie" title="Galoistheorie">Hauptsatz der Galoistheorie</a> formulieren:

<dl><dd>Die Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\mapsto G(L/Z)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\mapsto G(L/Z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db1194f6fefedfcb2907313efbe9a5acce5007cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.356ex; height:2.843ex;" alt="{\displaystyle Z\mapsto G(L/Z)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto L^{U}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\mapsto L^{U}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b22376e25ca812801528cce143adb3fb84f356ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.472ex; height:2.676ex;" alt="{\displaystyle U\mapsto L^{U}}" loading="lazy"> sind zueinander inverse Anti-Isomorphismen der beiden Verbände <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {Z}}(L/K)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Z}}(L/K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad92a4d98a38459570fde7930be24821f9526b7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.02ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {Z}}(L/K)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}(G)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {U}}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50f29fa93aa8b883eb89017ae8581e703de910ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.057ex; width:5.239ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {U}}(G)}" loading="lazy">, das heißt, sie sind zueinander inverse Bijektionen, welche die jeweiligen Inklusionen umkehren:

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\mathfrak {U}}(G)&\longleftrightarrow &{\mathfrak {Z}}(L/K)\\U&\longmapsto &L^{U}\\G(L/Z)&\leftarrowtail &Z\end{matrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bd60444a2f27283a7f941e0d8ed3eac97efc7a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-bottom: -0.333ex; width:25.797ex; height:9.676ex;" alt="{\displaystyle {\begin{matrix}{\mathfrak {U}}(G)&\longleftrightarrow &{\mathfrak {Z}}(L/K)\\U&\longmapsto &L^{U}\\G(L/Z)&\leftarrowtail &Z\end{matrix}}}" loading="lazy"></dd></dl></dd></dl>

Dass hierbei konjugierte Untergruppen auf konjugierte Fixpunktmengen (konjugierte Zwischenkörper) abgebildet werden, folgt mühelos aus den obigen allgemeinen Überlegungen. Also korrespondieren die in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> normalen (das heißt „unter der Konjugation invarianten“) Untergruppen (sprich: die Normalteiler) mit den über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> normalen Zwischenkörpern.

<div class="mw-heading mw-heading3"><h3 id="Moduln_und_Vektorräume">Moduln und Vektorräume</h3></div>

Ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-<a href="Modul_(Mathematik)#Moduln_über_einer_Gruppe" title="Modul (Mathematik)">(Links-)Modul</a> ist eine <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,+),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (M,+),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad2674954bae5dd4a95fbba75070952b9ca035cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.74ex; height:2.843ex;" alt="{\displaystyle (M,+),}" loading="lazy"> auf der eine Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> (von links) operiert, derart dass zusätzlich die (Links-)Operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright \colon G\times M\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright \colon G\times M\to M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d625a0a364db60ebf766b7f7411324374567557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.362ex; height:2.176ex;" alt="{\displaystyle \triangleright \colon G\times M\to M}" loading="lazy"> <a href="Vertr%C3%A4glichkeit_(Mathematik)" title="Verträglichkeit (Mathematik)">linksverträglich</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"> ist, d. h., es gelten

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\triangleright (x+y)=(g\triangleright x)+(g\triangleright y)\quad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35b985c25862416ebe7e77b087bdafba924f2f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.432ex; height:2.843ex;" alt="{\displaystyle g\triangleright (x+y)=(g\triangleright x)+(g\triangleright y)\quad }" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> und alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea304ca242a255b620d3dd16ec47f19efc2e7ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.802ex; height:2.509ex;" alt="{\displaystyle x,y\in M}" loading="lazy"></dd></dl>

Die Transformationen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{g\,\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta _{g\,\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc91646804c6d8d3963947a4819dfad912cc5b29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.604ex; height:2.843ex;" alt="{\displaystyle \vartheta _{g\,\triangleright }}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> bilden dann in der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {Aut} (M),\circ )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\operatorname {Aut} (M),\circ )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8efe8e08d92c37d5340878a8f6516c82db0f1e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.197ex; height:2.843ex;" alt="{\displaystyle (\operatorname {Aut} (M),\circ )}" loading="lazy"> der <a href="Automorphismus" title="Automorphismus">Automorphismen</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,+)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (M,+)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac85e31c15b188cc6674c500245dd710fc78d8ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.094ex; height:2.843ex;" alt="{\displaystyle (M,+)}" loading="lazy"> eine Untergruppe, und diese ist gerade das Bild des <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }\colon G\to \operatorname {Aut} (M);\;g\mapsto [\vartheta _{g\,\triangleright }:x\mapsto g\triangleright x]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }\colon G\to \operatorname {Aut} (M);\;g\mapsto [\vartheta _{g\,\triangleright }:x\mapsto g\triangleright x]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7cf2a9f33519e2639fd332a1789a9fe44c8dc50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.357ex; height:3.009ex;" alt="{\displaystyle \Theta _{\triangleright }\colon G\to \operatorname {Aut} (M);\;g\mapsto [\vartheta _{g\,\triangleright }:x\mapsto g\triangleright x]}" loading="lazy">.

Die Eigenschaft der „Verträglichkeit“ einer Gruppenoperation von links fließt beim Nachweis ein, dass es sich um einen Gruppenhomomorphismus handelt.

Im Falle einer Operation von rechts (also einem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Rechts-Modul) wäre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a88121f04a342077282bd71e1cfda9c6727c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \Theta _{\triangleright }}" loading="lazy"> ein „Gruppenantihomomorphismus“ oder aber – was auf dasselbe hinausläuft – erst dann ein Gruppenhomomorphismus, wenn anstelle der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ihre Gegengruppe als Definitionsbereich gewählt wird.

Ist insbesondere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \odot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e89e009eb8a8839c82aa5c76c15e9f2d67006276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }" loading="lazy"> die skalare Multiplikation eines <a href="Vektorraum" title="Vektorraum">Vektorraums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,\oplus ,\odot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (V,\oplus ,\odot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3cd271476aa9a75743edcc8f22ca88595f6a098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.281ex; height:2.843ex;" alt="{\displaystyle (V,\oplus ,\odot )}" loading="lazy"> über dem Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,+,\cdot ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K,+,\cdot ),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e0a49e7cf102c0793178d80a6334751a997dfc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.045ex; height:2.843ex;" alt="{\displaystyle (K,+,\cdot ),}" loading="lazy"> dann operiert die multiplikative Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K\setminus \{0\},\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K\setminus \{0\},\cdot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55e47aa90e60a65ec3b03b46c9e7cfd8b1487404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.238ex; height:2.843ex;" alt="{\displaystyle (K\setminus \{0\},\cdot )}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">.

Dies gilt auch für einen <a href="Schiefk%C3%B6rper" title="Schiefkörper">Schiefkörper</a>. Doch dabei ist auf die Seitigkeit zu achten: Bei Operation von links erhält man einen Links-Vektorraum, bei Operation von rechts einen Rechts-Vektorraum. Die wichtige Rolle der Seitigkeit wird auch beim Übergang zum <a href="Dualraum" title="Dualraum">Dualraum</a> eines Vektorraums deutlich, denn dabei wechselt die Seitigkeit: Der Dualraum eines Links-Vektorraumes ist ein Rechts-Vektorraum und umgekehrt. Erst der Bidualraum hat die ursprüngliche Seitigkeit – und ist im endlichdimensionalen Falle kanonisch isomorph mit dem Ausgangsvektorraum („<a href="Dualraum#Bidualraum" title="Dualraum">Dualitätssatz für Vektorrräume</a>“).<a href="#cite_note-20">[13]</a>

<div class="mw-heading mw-heading3"><h3 id="Kategorien">Kategorien</h3></div>

Ist allgemeiner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ein Objekt einer beliebigen <a href="Kategorientheorie" title="Kategorientheorie">Kategorie</a>, so kann eine <a href="Vertr%C3%A4glichkeit_(Mathematik)" title="Verträglichkeit (Mathematik)">strukturverträgliche</a> Operation einer (abstrakten) Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> definiert werden als ein Gruppenhomomorphismus

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\to \operatorname {Aut} X,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/beff8bdc98dd76a715dde98a69f0a4c925cbf3fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.395ex; height:2.509ex;" alt="{\displaystyle G\to \operatorname {Aut} X,}" loading="lazy"></dd></dl>

dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Aut} X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Aut} X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b395d70cfbf9736c8f3e649c20f704c6d547adb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.307ex; height:2.176ex;" alt="{\displaystyle \operatorname {Aut} X}" loading="lazy"> die Gruppe der Automorphismen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> im kategorientheoretischen Sinne. Die oben genannten Operationen von Gruppen auf Mengen oder abelschen Gruppen sind Spezialfälle.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="G-Raum" title="G-Raum">G-Raum</a> für stetige Gruppenwirkungen</li>

<li><a href="Lemma_von_Burnside" title="Lemma von Burnside">Lemma von Burnside</a></li>

<li><a href="Permutationsdarstellung" class="mw-redirect" title="Permutationsdarstellung">Permutationsdarstellung</a>, <a href="Permutationsgruppe" title="Permutationsgruppe">Permutationsgruppe</a></li>

<li><a href="Permutationsmatrix" title="Permutationsmatrix">Permutationsmatrix</a></li>

<li><a href="Darstellungstheorie" title="Darstellungstheorie">Darstellungstheorie</a></li>

<li><a href="Darstellungstheorie_endlicher_Gruppen" title="Darstellungstheorie endlicher Gruppen">Darstellungstheorie endlicher Gruppen</a></li>

<li><a href="Darstellungstheorie_kompakter_Gruppen" title="Darstellungstheorie kompakter Gruppen">Darstellungstheorie kompakter Gruppen</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Group_actions?uselang=de">Commons: Group actions</a> – Sammlung von Bildern, Videos und Audiodateien</div>

<ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/GroupAction.html">Gruppenoperation bei MathWorld</a> (englisch)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Von Lateinisch coniugare: „(zu einem Paar) verbinden, verknüpfen; verheiraten“: Konjugierte Elemente sind mit Hilfe der Gruppenoperation miteinander verknüpft, nämlich durch die Operation des geeigneten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ae77eeb0200a3b0e26ff4b251fb845ca1b385c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:3.009ex;" alt="{\displaystyle g^{-1}}" loading="lazy">.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Beachte hierzu unter den <a href="#Beispiele">Beispielen</a> den Abschnitt <a href="#Operation_durch_Konjugation">Operation durch Konjugation</a>.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> In der Literatur scheint der Begriff „Stabilisator einer Menge“ nicht üblich zu sein, möglicherweise um Verwechslungen mit der Fixgruppe von vornherein zu vermeiden. Zudem wird sich weiter unten zeigen, dass er mit dem Normalisator der Fixgruppe übereinstimmt, so dass die Einführung einer potenziell verwirrenden Begrifflichkeit verzichtbar ist.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Beachte, dass es der Transitivität wegen eine Bijektion zwischen einer Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\subseteq G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I\subseteq G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bc29cc376828a933690ed54d4bfd03486ecc48d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.097ex; height:2.343ex;" alt="{\displaystyle I\subseteq G}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> gibt: Zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> gibt es ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5bc828acb4ef05849654ac86b14bc94ebac17c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.47ex; height:2.176ex;" alt="{\displaystyle i\in G}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\triangleright x_{0}=x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\triangleright x_{0}=x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/917f2e229aeae5f673a5aa4a1e4be0a36488d803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.809ex; height:2.509ex;" alt="{\displaystyle i\triangleright x_{0}=x}" loading="lazy">.

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> Man beachte die obige Feststellung, dass eine Gruppenoperation „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleright }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \triangleright }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23951eaf8dba0ab46e9884c6c0f717558acc2195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle \triangleright }" loading="lazy">“ von der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54a87abf331634c8962ef14c4c5ec41f94fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}" loading="lazy"> auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> mit einem Gruppenhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\triangleright }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\triangleright }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de8173e7a8150f7ff4235f2b6ab2aeeba74a07be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{\triangleright }}" loading="lazy"> von dieser Gruppe in die symmetrische Gruppe (Permutationsgruppe) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f4a9bd46fe22dca3aeaa37d032281966a79a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> gleichgesetzt werden kann: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\triangleright }\colon G\to \operatorname {Sym} (X),\;g\mapsto [\varphi _{\triangleright }(g)\colon x\mapsto g\triangleright x]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\triangleright }\colon G\to \operatorname {Sym} (X),\;g\mapsto [\varphi _{\triangleright }(g)\colon x\mapsto g\triangleright x]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3655c1d00e277c70cd8a85707be16fb5d90dcf5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.115ex; height:2.843ex;" alt="{\displaystyle \varphi _{\triangleright }\colon G\to \operatorname {Sym} (X),\;g\mapsto [\varphi _{\triangleright }(g)\colon x\mapsto g\triangleright x]}" loading="lazy">. Genau dieser Fall liegt hier vor, indem man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/753cb013502f93e665478318c42820c573d7dc13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.905ex; height:2.176ex;" alt="{\displaystyle X=G}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\triangleright }(g)=\varphi _{g}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\triangleright }(g)=\varphi _{g}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a079b1742e639bfa84bd2e4e564d5db5caddf9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.14ex; height:3.009ex;" alt="{\displaystyle \varphi _{\triangleright }(g)=\varphi _{g}}" loading="lazy"> setzt. Dabei handelt es sich bei den Permutationen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/753cb013502f93e665478318c42820c573d7dc13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.905ex; height:2.176ex;" alt="{\displaystyle X=G}" loading="lazy"> sogar um Automorphismen.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">↑</a> Die Separabilität der Erweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> ergibt sich daraus, dass der Grundkörper als ein Fixkörper gewählt wurde.

</li>

<li id="cite_note-19"><a href="#cite_ref-19">↑</a> Diese Formulierung kommt der <a href="Galoistheorie#Klassischer_Ansatz" title="Galoistheorie">historischen Perspektive</a> auf die Theorie höherer Gleichungen – auf die ursprüngliche historische Fragstellung der Algebra – nahe, welche schließlich im <a href="Satz_von_Abel-Ruffini" title="Satz von Abel-Ruffini">Satz von Abel-Ruffini</a> (1799, 1824) und in der <a href="Galois-Theorie" class="mw-redirect" title="Galois-Theorie">Galois-Theorie</a> gipfelte: Man betrachtete die Permutationsgruppe der Nullstellenmenge eines Polynoms und nutzte Symmetrien (Stichworte: <a href="Lagrange-Resolvente" title="Lagrange-Resolvente">Lagrange-Resolvente</a> und <a href="Elementarsymmetrisches_Polynom#Eigenschaften" title="Elementarsymmetrisches Polynom">Hauptsatz über elementarsymmetrische Funktionen</a>). Der <a href="Galoistheorie#Moderner_Ansatz" title="Galoistheorie">moderne Ansatz</a> hingegen stellt die Automorphismengruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> als Transformationsgruppe auf dem Erweiterungskörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> in den Mittelpunkt der Betrachtungen.

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <cite style="font-style:italic">Fundamentalbereich</cite>. In: Guido Walz (Hrsg.): <cite style="font-style:italic">Lexikon der Mathematik</cite>. 1. Auflage. Spektrum Akademischer Verlag, Mannheim/Heidelberg 2000, ISBN 3-8274-0439-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Fundamentalbereich&rft.btitle=Lexikon+der+Mathematik&rft.date=2000&rft.edition=1.&rft.genre=book&rft.isbn=3827404398&rft.place=Mannheim%2FHeidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Nieper-Wißkirchen: Galoissche Theorie. Universität Augsburg (2013), Folgerung 4.11, S. 133 (<style data-mw-deduplicate="TemplateStyles:r261891140">

/* start https://de.wikipedia.org/ */

.mw-parser-output .webarchiv-memento a{color:inherit}

/* end https://de.wikipedia.org/ */

</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20190715130507/https://www.math.uni-augsburg.de/de/prof/alg/Downloads/Einfuehrung-in-die-Algebra.pdf">online</a> (<a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a> vom 15. Juli 2019 im <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>)).

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Vgl. auch <a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Bartel Leendert van der Waerden</a>: <cite style="font-style:italic">Algebra I</cite>. unter Benutzung von Vorlesungen von E. Artin und E. Noether. (= <cite style="font-style:italic">Heidelberger Taschenbücher</cite>. Band 12). 8. Auflage. 1971, ISBN 3-540-03561-3, Kapitel VI Körpertheorie, § 39 Einfache Körpererweiterungen, Satz oben (vierter Absatz) auf Seite 117 (in Verbindung mit dem <a href="Satz_vom_primitiven_Element" title="Satz vom primitiven Element">Satz vom primitiven Element</a> für den <a href="Zerf%C3%A4llungsk%C3%B6rper" title="Zerfällungskörper">Zerfällungskörper</a> des gegebenen Polynoms).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Kapitel+VI+K%C3%B6rpertheorie%27%27%2C+%C2%A7+39+%27%27Einfache+K%C3%B6rpererweiterungen%2C+Satz+oben+%28vierter+Absatz%29+auf+Seite+117+%28in+Verbindung+mit+dem+Satz+vom+primitiven+Element+f%C3%BCr+den+Zerf%C3%A4llungsk%C3%B6rper+des+gegebenen+Polynoms%29&rft.au=Bartel+Leendert+van+der+Waerden&rft.btitle=Algebra+I&rft.date=1971&rft.edition=8.&rft.genre=bookitem&rft.isbn=3540035613&rft.series=Heidelberger+Taschenb%C3%BCcher" style="display:none">

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> <a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite> (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959, Abschnitt II Galoissche Theorie. Reintranszendente Körper, § 11 und 12., S. 37 ff. (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Abschnitt+II+Galoissche+Theorie.+Reintranszendente+K%C3%B6rper%2C+%C2%A7+11+und+12.&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=bookitem&rft.pages=37+ff.&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none">

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> <a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite> (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959, Abschnitt I. Gruppentheorie, § 7 Permutationsgruppen, Satz 3, S. 20 oben (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Abschnitt+I.+Gruppentheorie%2C+%C2%A7+7+Permutationsgruppen%2C+Satz+3&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=bookitem&rft.pages=20+oben&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> <a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Bartel Leendert van der Waerden</a>: <cite style="font-style:italic">Algebra I</cite>. unter Benutzung von Vorlesungen von E. Artin und E. Noether. (= <cite style="font-style:italic">Heidelberger Taschenbücher</cite>. Band 12). 8. Auflage. 1971, ISBN 3-540-03561-3, Kapitel VI Fortsetzung der Gruppentheorie, § 56 Transitivität und Primitivität, Seite 165ff.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Kapitel+VI+Fortsetzung+der+Gruppentheorie%27%27%2C+%C2%A7+56+%27%27Transitivit%C3%A4t+und+Primitivit%C3%A4t%2C+Seite+165ff.&rft.au=Bartel+Leendert+van+der+Waerden&rft.btitle=Algebra+I&rft.date=1971&rft.edition=8&rft.genre=bookitem&rft.isbn=3540035613&rft.series=Heidelberger+Taschenb%C3%BCcher" style="display:none">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Vgl.

<ul><li><a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite> (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959, I. Gruppentheorie § 7 Permutationsgruppen., S. 26 (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=I.+Gruppentheorie+%C2%A7+7+Permutationsgruppen.&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=bookitem&rft.pages=26&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none"> </li>

<li><a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Bartel Leendert van der Waerden</a>: <cite style="font-style:italic">Algebra II</cite>. Unter Benutzung von Vorlesungen von E. Artin und E. Noether (= <cite style="font-style:italic">Heidelberger Taschenbücher</cite>. Band 23). 5. Auflage. Springer-Verlag, Heidelberg Berlin 1967, ISBN 3-540-03869-8, XIV. Kapitel, § 108 Darstellungen endlicher Gruppen, S. 97 ff. (Originaltitel der ersten Auflage: Moderne Algebra).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=XIV.+Kapitel%2C+%C2%A7+108+Darstellungen+endlicher+Gruppen&rft.au=Bartel+Leendert+van+der+Waerden&rft.btitle=Algebra+II&rft.date=1967&rft.edition=5.&rft.genre=bookitem&rft.isbn=3540038698&rft.pages=97+ff.&rft.place=Heidelberg+Berlin&rft.pub=Springer-Verlag&rft.series=Heidelberger+Taschenb%C3%BCcher" style="display:none"> </li>

<li><a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Linear Representations of Finite Groups</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Graduate Texts in Mathematics</cite>. Band 42). 3. Auflage. Springer-Verlag, New York Heidelberg Berlin 1977 (englisch, 170 S., französisch: <cite class="lang" lang="fr" dir="auto" style="font-style:italic">Représentations linéaires des groupes finis</cite>. Paris 1971.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.au=Jean-Pierre+Serre&rft.btitle=Linear+Representations+of+Finite+Groups&rft.date=1977&rft.edition=3.&rft.genre=book&rft.place=New+York+Heidelberg+Berlin&rft.pub=Springer-Verlag&rft.series=Graduate+Texts+in+Mathematics" style="display:none"> </li></ul>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> Siehe <a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite> (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959, I. Gruppentheorie § 7 Permutationsgruppen., S. 26/27 (Seitenumbruch) (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=I.+Gruppentheorie+%C2%A7+7+Permutationsgruppen.&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=bookitem&rft.pages=26%2F27+%28Seitenumbruch%29&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> Siehe <a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite>. Band II (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959, Abschnitt I Gruppentheorie, § 9 Die alternierende Gruppe, Satz 5., S. 31 (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Abschnitt+I+Gruppentheorie%27%27%2C+%C2%A7+9+%27%27Die+alternierende+Gruppe%2C+Satz+5.&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=bookitem&rft.pages=31&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none">

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> Siehe <a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite> (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959, I. Gruppentheorie § 7 Permutationsgruppen Satz 4, Seite 26. (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=I.+Gruppentheorie+%C2%A7+7+Permutationsgruppen+Satz+4%2C+Seite+26.&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=bookitem&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none">

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Siehe <a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite> (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959, I. Gruppentheorie § 7 Permutationsgruppen., S. 24 ff. (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=I.+Gruppentheorie+%C2%A7+7+Permutationsgruppen.&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=bookitem&rft.pages=24ff.&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none">

</li>

<li id="cite_note-18"><a href="#cite_ref-18">↑</a> <a href="Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: <cite style="font-style:italic">Algebra</cite>. 10. Auflage. Springer, 2023, ISBN 978-3-662-67463-5, Kapitel 4 Galois-Theorie, 4.1 Galois-Erweiterungen, Satz 4, S. 190, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-67464-2">10.1007/978-3-662-67464-2</a> (508 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Kapitel+4+Galois-Theorie%2C+4.1+Galois-Erweiterungen%2C+Satz+4&rft.au=Siegfried+Bosch&rft.btitle=Algebra&rft.date=2023&rft.doi=10.1007%2F978-3-662-67464-2&rft.edition=10.&rft.genre=bookitem&rft.isbn=9783662674635&rft.pages=190&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-20"><a href="#cite_ref-20">↑</a> <a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Bartel Leendert van der Waerden</a>: <cite style="font-style:italic">Algebra I</cite>. unter Benutzung von Vorlesungen von E. Artin und E. Noether. (= <cite style="font-style:italic">Heidelberger Taschenbücher</cite>. Band 12). 8.  Auflage. 1971, ISBN 3-540-03561-3, Kapitel IVI Vektorräume und Tensorräume, § 21 Der duale Vektorraum., S. 68 f. (272 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Kapitel+IVI+Vektorr%C3%A4ume+und+Tensorr%C3%A4ume%27%27%2C+%C2%A7+21+%27%27Der+duale+Vektorraum.&rft.au=Bartel+Leendert+van+der+Waerden&rft.btitle=Algebra+I&rft.date=1971&rft.edition=8.+Auflage&rft.genre=bookitem&rft.isbn=3540035613&rft.pages=68f.&rft.series=Heidelberger+Taschenb%C3%BCcher" style="display:none">

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><cite style="font-style:italic">Fundamentalbereich</cite>. In: Guido Walz (Hrsg.): <cite style="font-style:italic">Lexikon der Mathematik</cite>. 1. Auflage. Spektrum Akademischer Verlag, Mannheim/Heidelberg 2000, ISBN 3-8274-0439-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Fundamentalbereich&rft.btitle=Lexikon+der+Mathematik&rft.date=2000&rft.edition=1.&rft.genre=book&rft.isbn=3827404398&rft.place=Mannheim%2FHeidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </li>

<li>Nieper-Wißkirchen: Galoissche Theorie. Universität Augsburg (2013), Folgerung 4.11, S. 133 (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20190715130507/https://www.math.uni-augsburg.de/de/prof/alg/Downloads/Einfuehrung-in-die-Algebra.pdf">online</a> (<a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a> vom 15. Juli 2019 im <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>))</li>

<li><a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a>: <cite style="font-style:italic">Elementare und klassische Algebra</cite> (= <cite style="font-style:italic">Sammlung Göschen</cite>. Band 933). Walter de Gruyter & Co, Berlin 1959 (132 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.au=Wolfgang+Krull&rft.btitle=Elementare+und+klassische+Algebra&rft.date=1959&rft.genre=book&rft.place=Berlin&rft.pub=Walter+de+Gruyter+%26+Co&rft.series=Sammlung+G%C3%B6schen" style="display:none"> </li>

<li><a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Bartel Leendert van der Waerden</a>: <cite style="font-style:italic">Algebra I</cite>. unter Benutzung von Vorlesungen von E. Artin und E. Noether. (= <cite style="font-style:italic">Heidelberger Taschenbücher</cite>. Band 12). 8.  Auflage. 1971, ISBN 3-540-03561-3.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.au=Bartel+Leendert+van+der+Waerden&rft.btitle=Algebra+I&rft.date=1971&rft.edition=8.+Auflage&rft.genre=book&rft.isbn=3540035613&rft.series=Heidelberger+Taschenb%C3%BCcher" style="display:none"> </li>

<li><a href="Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: <cite style="font-style:italic">Algebra</cite>. 10. Auflage. Springer, 2023, ISBN 978-3-662-67463-5, Kapitel 4. Galois-Theorie, 4.1 Galois-Erweiterungen, Satz 4, S. 190, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-67464-2">10.1007/978-3-662-67464-2</a> (508 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Gruppenoperation&rft.atitle=Kapitel+4.+Galois-Theorie%2C+4.1+Galois-Erweiterungen%2C+Satz+4&rft.au=Siegfried+Bosch&rft.btitle=Algebra&rft.date=2023&rft.doi=10.1007%2F978-3-662-67464-2&rft.edition=10.&rft.genre=bookitem&rft.isbn=9783662674635&rft.pages=190&rft.pub=Springer" style="display:none"> </li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2026-01-10" href="https://de.wikipedia.org/wiki/?title=Gruppenoperation&oldid=263205193">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>