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<title>Normalisator</title>

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

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<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">Der Normalisator ist ein Begriff aus dem <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebiet</a> der <a href="Gruppentheorie" title="Gruppentheorie">Gruppentheorie</a>.

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

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

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

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</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 <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</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"> eine nichtleere Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</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">. Der Normalisator von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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

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

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</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 definiert als

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<annotation encoding="application/x-tex">{\displaystyle N_{G}(U):=\left\{g\in G\mid gUg^{-1}=U\right\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77de2315674764de1b44dd527ffff773a808484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.389ex; height:3.343ex;" alt="{\displaystyle N_{G}(U):=\left\{g\in G\mid gUg^{-1}=U\right\}}" loading="lazy">.</dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gUg^{-1}=\left\{gug^{-1}\mid u\in U\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle gUg^{-1}=\left\{gug^{-1}\mid u\in U\right\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a4a13dc04185fd822f88a0531deb027b187a5d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.945ex; height:3.343ex;" alt="{\displaystyle gUg^{-1}=\left\{gug^{-1}\mid u\in U\right\}}" loading="lazy">, entsprechend der Definition des <a href="Komplexprodukt" title="Komplexprodukt">Komplexproduktes</a>.<a href="#cite_note-meyberg186-1">[1]</a><a href="#cite_note-robinson-2">[2]</a>

Mit anderen Worten: Der Normalisator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{G}(U)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fee67f3cd674557c68f704f30908adcc238ee66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle N_{G}(U)}" loading="lazy"> besteht aus denjenigen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

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

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</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 die gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</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 <a href="Konjugation_(Gruppentheorie)" title="Konjugation (Gruppentheorie)">Konjugation</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</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"> invariant ist. (Man sagt, dass diese Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</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"> normalisieren.)

Man beachte, dass lediglich gefordert wird, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</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"> als Ganzes festbleibt, im Allgemeinen gilt also für einzelne Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65c3e0203fca1c3b6c1982520132b9f057ed5d88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle u\in U}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in N_{G}(U)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6df22b18233e1840b6755a3930455e30ac0f7526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.939ex; height:2.843ex;" alt="{\displaystyle g\in N_{G}(U)}" loading="lazy"> durchaus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gug^{-1}\neq u}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7345eb4058163d8e5c46967a6136a992f34b760b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.325ex; height:3.176ex;" alt="{\displaystyle gug^{-1}\neq u}" loading="lazy">; es gilt aber stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gug^{-1}\in U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0746f8ad1c8cad971539b6a37e93318411d3ea93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.52ex; height:3.009ex;" alt="{\displaystyle gug^{-1}\in U}" loading="lazy">.

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

<ul><li>Der Normalisator ist eine Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</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="#cite_note-meyberg187-3">[3]</a></li>

<li>Der <a href="Index_(Gruppentheorie)" title="Index (Gruppentheorie)">Index</a> des Normalisators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{G}(U)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fee67f3cd674557c68f704f30908adcc238ee66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle N_{G}(U)}" loading="lazy"> liefert die Anzahl der unterschiedlichen Konjugierten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gUg^{-1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85b1ef378f287793ba06ea30d84544ba6e548b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.35ex; height:3.009ex;" alt="{\displaystyle gUg^{-1}}" loading="lazy"> der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</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">, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\{gUg^{-1}\mid g\in G\}|=[G:N_{G}(U)]}">

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<annotation encoding="application/x-tex">{\displaystyle |\{gUg^{-1}\mid g\in G\}|=[G:N_{G}(U)]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64363b24011d9cc88d8e63052359c14f55e03067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.827ex; height:3.176ex;" alt="{\displaystyle |\{gUg^{-1}\mid g\in G\}|=[G:N_{G}(U)]}" loading="lazy">.</li>

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

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

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</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"> ist stets <a href="Normalteiler" title="Normalteiler">Normalteiler</a> in ihrem Normalisator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{G}(U)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fee67f3cd674557c68f704f30908adcc238ee66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle N_{G}(U)}" loading="lazy">.<a href="#cite_note-meyberg187-3">[3]</a> Genauer: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{G}(U)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fee67f3cd674557c68f704f30908adcc238ee66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle N_{G}(U)}" loading="lazy"> ist die bezüglich <a href="Teilmenge" title="Teilmenge">Inklusion</a> größte <a href="Untergruppe" title="Untergruppe">Untergruppe</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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

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

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</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"> Normalteiler ist.</li>

<li>Eine Untergruppe ist genau dann Normalteiler in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</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 ihr Normalisator ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</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.<a href="#cite_note-meyberg187-3">[3]</a></li>

<li>Man kann den Normalisator auch wie folgt einführen: Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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

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

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</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 der <a href="Potenzmenge" title="Potenzmenge">Potenzmenge</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</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 <a href="Konjugation_(Gruppentheorie)" title="Konjugation (Gruppentheorie)">Konjugation</a> <a href="Gruppenoperation" title="Gruppenoperation">operieren</a>. Dann ist der <a href="Gruppenoperation#Stabilisator" title="Gruppenoperation">Stabilisator</a> dieser Operation für eine gegebene Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</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"> gerade der Normalisator dieser Teilmenge.</li></ul>

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

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

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

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</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="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">Gruppe der invertierbaren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Matrizen</a> (mit reellen Einträgen) für eine natürliche Zahl <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">. Weiter sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

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</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"> die Untergruppe der <a href="Diagonalmatrix" title="Diagonalmatrix">Diagonalmatrizen</a>. Dann ist der Normalisator von <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"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</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 Gruppe der Matrizen, bei denen in jeder Zeile und in jeder Spalte genau ein Eintrag ungleich null ist. Der <a href="Faktorgruppe" title="Faktorgruppe">Quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{G}(U)/U}">

<msub>

</mrow>

</msub>

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</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5585edd2d2cab11885d0e150136d778fc4b20202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.927ex; height:2.843ex;" alt="{\displaystyle N_{G}(U)/U}" loading="lazy"> ist <a href="Isomorph" class="mw-redirect" title="Isomorph">isomorph</a> zur <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrischen Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}" loading="lazy">.<a href="#cite_note-Procesi-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Verwandte_Begriffe">Verwandte Begriffe</h2></div>

Fordert man, dass <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"> elementweise invariant unter der Konjugation mit Gruppenelementen ist, erhält man den stärkeren Begriff des <a href="Zentralisator" title="Zentralisator">Zentralisators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{G}(U)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bd5ec19e860a864f6b49fd9df173813a95cae85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.703ex; height:2.843ex;" alt="{\displaystyle Z_{G}(U)}" loading="lazy">. Der Zentralisator ist ein Normalteiler im jeweiligen Normalisator.<a href="#cite_note-robinson-2">[2]</a>

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

Kurt Meyberg: <cite style="font-style:italic">Algebra Teil 1</cite>. Karl Hanser Verlag, 1980, ISBN 3-446-13079-9, S. 52, Definition 1.8.6.<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:Normalisator&rft.au=Kurt+Meyberg&rft.btitle=Algebra+Teil+1&rft.date=1980&rft.genre=book&rft.isbn=3446130799&rft.pages=52&rft.pub=Karl+Hanser+Verlag" style="display:none">

</li>

Derek J. S. Robinson: <cite class="lang" lang="en" dir="auto" style="font-style:italic">A Course in the Theory of Groups</cite>. Springer, 1996, ISBN 1-4612-6443-X, S. 38 (englisch).<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:Normalisator&rft.au=Derek+J.+S.+Robinson&rft.btitle=A+Course+in+the+Theory+of+Groups&rft.date=1996&rft.genre=book&rft.isbn=146126443X&rft.pages=38&rft.pub=Springer" style="display:none">

</li>

Kurt Meyberg: <cite style="font-style:italic">Algebra Teil 1</cite>. Karl Hanser Verlag, 1980, ISBN 3-446-13079-9, S. 53, Satz 1.8.7.<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:Normalisator&rft.au=Kurt+Meyberg&rft.btitle=Algebra+Teil+1&rft.date=1980&rft.genre=book&rft.isbn=3446130799&rft.pages=53&rft.pub=Karl+Hanser+Verlag" style="display:none">

</li>

Claudio Procesi: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Lie Groups</cite>. An Approach through Invariants and Representations. Springer, 2007, ISBN 978-0-387-26040-2, S. 218, Kap. 4.8 Representations of Linearly Reductive Groups (englisch).<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:Normalisator&rft.au=Claudio+Procesi&rft.btitle=Lie+Groups&rft.date=2007&rft.genre=book&rft.isbn=9780387260402&rft.pages=218&rft.pub=Springer" style="display:none">

</li>

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