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<title>Subquotient</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Subquotient</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">In den <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebieten</a> der <a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a> und der <a href="Abstrakte_Algebra" title="Abstrakte Algebra">Abstrakten Algebra</a> versteht man unter einem Subquotienten ein Quotientenobjekt eines Unterobjekts.

In der Sprache der <a href="Gruppentheorie" title="Gruppentheorie">Gruppentheorie</a> ist ein Unterobjekt eine <a href="Untergruppe" title="Untergruppe">Untergruppe</a> und ein Quotientenobjekt eine <a href="Quotientengruppe" class="mw-redirect" title="Quotientengruppe">Quotientengruppe</a> (auch <a href="Faktorgruppe" title="Faktorgruppe">Faktorgruppe</a> genannt): Damit ist ein Subquotient einer Gruppe <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="Isomorph" class="mw-redirect" title="Isomorph">isomorph</a> zum Bild einer 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"> unter einem <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismus</a>. Und wie beim Begriff der Untergruppe werden <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"> selbst und die einelementige Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5acdcac635f883f8b4f0a01aa03b16b22f23b124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}" loading="lazy"> als triviale Subquotienten angesehen.

Der Begriff Subquotient findet Anwendung u. a. bei der Klassifikation der <a href="Endliche_einfache_Gruppe" title="Endliche einfache Gruppe">endlichen einfachen Gruppen</a>, insbesondere bei den <a href="Sporadische_Gruppe" title="Sporadische Gruppe">sporadischen Gruppen</a>.

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

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

Ist <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, <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_/eb734a37dd21ce173a46342d1cc64c92/76634fad5818a777669a77cd8c86d1d816e4c402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}" loading="lazy"> 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"> und <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_/eb734a37dd21ce173a46342d1cc64c92/5a37ec377e9bb29c7dd95a844c1b230fbbebea75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.964ex; height:2.509ex;" alt="{\displaystyle G''}" loading="lazy"> ein <a href="Normalteiler" title="Normalteiler">Normalteiler</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_/eb734a37dd21ce173a46342d1cc64c92/76634fad5818a777669a77cd8c86d1d816e4c402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}" loading="lazy">, in Zeichen

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3e500d36a4bbc755dc326c20ba40ec7c6f74fbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.146ex; height:2.843ex;" alt="{\displaystyle G\geq G'\vartriangleright G'',}" loading="lazy"></dd></dl>

dann nennt man die Faktorgruppe (Quotientengruppe) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:=G'/G''}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5cd0bfc3b39304f014109cd4fa636259267b016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.447ex; height:3.009ex;" alt="{\displaystyle H:=G'/G''}" loading="lazy"> einen Subquotienten 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 Literatur über sporadische Gruppen finden sich Formulierungen wie

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

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

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</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"><a href="#cite_note-1">[1]</a></li>

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

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</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 involviert 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"><a href="#cite_note-2">[2]</a></li></ol>

für denselben <a href="Sachverhalt" title="Sachverhalt">Sachverhalt</a>.

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

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

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

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</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 <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring mit Einselement</a>. Bei den <a href="Modul_(Mathematik)" title="Modul (Mathematik)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</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">-Moduln</a> gibt es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</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">-<a href="Untermodul" title="Untermodul">Untermoduln</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</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">-<a href="Quotientenmodul" title="Quotientenmodul">Quotientenmoduln</a> (Faktormoduln). Ganz analog wie bei den Gruppen sind die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</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">-Subquotienten definiert.

Die Begriffsbildung gilt auch bei nicht-kommutativem Ring und links/rechts-seitigen Moduln über diesem Ring.

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

<ul><li>Die <a href="Endliche_einfache_Gruppe" title="Endliche einfache Gruppe">einfache</a> <a href="A5_(Gruppe)" title="A5 (Gruppe)">alternierende Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation>

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</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"> vom Grad 5 hat die nicht-einfache <a href="A4_(Gruppe)" title="A4 (Gruppe)">alternierende Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{4}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e3eaac426165e199a4a745aa46bb8028fed100d.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_{4}}" loading="lazy"> vom Grad 4 zum Subquotienten (zur Untergruppe).</li>

<li>Ein Unterobjekt 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"> wie auch ein (homomorphes) Bild 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"> ist ein Subquotient 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/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"></li></ul>

<ul><li>Die <a href="Subnormalreihe" class="mw-redirect" title="Subnormalreihe">Faktoren einer Subnormalreihe</a> sind Subquotienten.</li></ul>

<ul><li>Das <a href="Schmetterlingslemma" class="mw-redirect" title="Schmetterlingslemma">Schmetterlingslemma</a> trifft eine Aussage über die Isomorphie gewisser Subquotienten.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Endliche_Objekte">Endliche Objekte</h2></div>

Haben alle Objekte endliche <a href="M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Kardinalitäten</a>, dann gibt es Formeln, die diese mit <a href="Index_(Gruppentheorie)" title="Index (Gruppentheorie)">Indices</a> in Beziehung bringen, siehe zum Beispiel den <a href="Satz_von_Lagrange" title="Satz von Lagrange">Satz von Lagrange</a>. Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H|=[H:1]=[G':G'']}">

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<annotation encoding="application/x-tex">{\displaystyle |H|=[H:1]=[G':G'']}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/754e5b4e41f347dc4fc819e1d2c1a9425990b773.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.717ex; height:3.009ex;" alt="{\displaystyle |H|=[H:1]=[G':G'']}" loading="lazy"> gilt mit obigen Bezeichnungen

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<annotation encoding="application/x-tex">{\displaystyle |G|=[G:1]=[G:G']\cdot [G':G'']\cdot [G'':1]=[G:G']\cdot |H|\cdot |G''|,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/302611e6717048e84db8c6419b7f15dcf7cdab8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.815ex; height:3.009ex;" alt="{\displaystyle |G|=[G:1]=[G:G']\cdot [G':G'']\cdot [G'':1]=[G:G']\cdot |H|\cdot |G''|,}" loading="lazy"></dd></dl>

und ist insbesondere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H|}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31c3c97e64ad558dcc09e9ff232ee0f4d447bac1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.357ex; height:2.843ex;" alt="{\displaystyle |H|}" loading="lazy"> ein Teiler 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_/eb734a37dd21ce173a46342d1cc64c92/8258bc41edeb87bfbc8cba8367f29838c0eddc1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.12ex; height:2.843ex;" alt="{\displaystyle |G|}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H|\leq |G|.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/015c704bd9249733eb1a0ac4bc5a560d35cc8217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.223ex; height:2.843ex;" alt="{\displaystyle |H|\leq |G|.}" loading="lazy">

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

Für endliche Objekte ist die Relation »ist Subquotient von« eine Ordnungsrelation, und zwar eine <a href="Halbordnung" class="mw-redirect" title="Halbordnung">Halbordnung</a>.

<div class="mw-heading mw-heading3"><h3 id="Reflexivität"><a href="Reflexive_Relation" title="Reflexive Relation">Reflexivität</a></h3></div>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93750259fbff9d764e4a8c2ff292baa60ecf84f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:2.843ex;" alt="{\displaystyle G/\{1\}}" loading="lazy"> ist Subquotient 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">.

<div class="mw-heading mw-heading3"><h3 id="Antisymmetrie"><a href="Antisymmetrische_Relation" title="Antisymmetrische Relation">Antisymmetrie</a></h3></div>

Sind zwei Objekte Subquotienten voneinander, so sind sie <a href="Isomorphismus" title="Isomorphismus">isomorph</a>.

<dl><dt>Beweis</dt></dl>

Die Wechselbeziehung zwischen <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 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"> lässt sich wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H|\leq |G|\leq |H|}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/391d9e310ba729b05b2c60c9d96beff193787423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.032ex; height:2.843ex;" alt="{\displaystyle |H|\leq |G|\leq |H|}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H|=|G|}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/791736612f9355979daecc462466d51892d0cc78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.576ex; height:2.843ex;" alt="{\displaystyle |H|=|G|}" loading="lazy">, nur aufrechterhalten mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [G:G']=1,[G'':1]=1}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [G:G']=1,[G'':1]=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/959f79b0c0a75930e7582919f25621b565cc9b40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.482ex; height:3.009ex;" alt="{\displaystyle [G:G']=1,[G'':1]=1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H:H']=1,[H'':1]=1}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [H:H']=1,[H'':1]=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a913f72f99c069735c6b39738f3ab1913fefe73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.272ex; height:3.009ex;" alt="{\displaystyle [H:H']=1,[H'':1]=1}" loading="lazy">, woraus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cong H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f74c426546451af44e2fbc290e06cc084929e6b3.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 G\cong H}" loading="lazy"> folgt.

<div class="mw-heading mw-heading3"><h3 id="Transitivität"><a href="Transitive_Relation" title="Transitive Relation">Transitivität</a></h3></div>

Subquotienten von Subquotienten sind Subquotienten.

<dl><dt>Beweis für Gruppen</dt></dl>

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

<msup>

</msup>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5cd0bfc3b39304f014109cd4fa636259267b016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.447ex; height:3.009ex;" alt="{\displaystyle H:=G'/G''}" loading="lazy"> Subquotient 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 \varphi \colon G'\to H}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon G'\to H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f489d4451917bb8b2720f7c443a17a2521ebf2c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.743ex; height:3.009ex;" alt="{\displaystyle \varphi \colon G'\to H}" loading="lazy"> der kanonische Homomorphismus. Ist nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\geq H'\vartriangleright H'',}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25d0c29cf7e43c8eaccf2779b49378eaadfebd55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.936ex; height:2.843ex;" alt="{\displaystyle H\geq H'\vartriangleright H'',}" loading="lazy"> also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H'/H''}">

<msup>

</msup>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e133f56a94114a159d33fc982427bbf9512830c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.191ex; height:3.009ex;" alt="{\displaystyle H'/H''}" loading="lazy"> Subquotient 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/ef601e1519093ba6c2944b945882c119f990e704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle H,}" loading="lazy"> dann sind die durch senkrechte Pfeile (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4618f22b0f780805eb94bb407578d9bc9487947a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }" loading="lazy">) gekennzeichneten Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e0b4482cc623f9ed3521235eeee6d29e724db0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.225ex; height:3.009ex;" alt="{\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''}" loading="lazy">

</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"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.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 \geq }" loading="lazy"></td>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76634fad5818a777669a77cd8c86d1d816e4c402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.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 \geq }" loading="lazy"></td>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{-1}(H')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05bb9679df63557dd17ce412bfe31527544c1ba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.45ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}(H')}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.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 \geq }" loading="lazy"></td>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{-1}(H'')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abdbf35244e79141c39480bfd0c870686ef8c841.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.903ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}(H'')}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0d20eca58300d3e12e1a099bd96e616d32bd426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \vartriangleright }" loading="lazy"></td>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a37ec377e9bb29c7dd95a844c1b230fbbebea75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.964ex; height:2.509ex;" alt="{\displaystyle G''}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dff1165f3a61965dc4d8c380a2de6dfab0e528cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.425ex; height:2.176ex;" alt="{\displaystyle \varphi \!:}" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33194d9036970c4980c32f4293bec52fa4a3edc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:1.55ex; height:4.176ex;" alt="{\displaystyle {\Big \downarrow }}" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33194d9036970c4980c32f4293bec52fa4a3edc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:1.55ex; height:4.176ex;" alt="{\displaystyle {\Big \downarrow }}" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33194d9036970c4980c32f4293bec52fa4a3edc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:1.55ex; height:4.176ex;" alt="{\displaystyle {\Big \downarrow }}" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33194d9036970c4980c32f4293bec52fa4a3edc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:1.55ex; height:4.176ex;" alt="{\displaystyle {\Big \downarrow }}" loading="lazy">

</td></tr>

<tr>

</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"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.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 \geq }" loading="lazy"></td>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/575e08b1574dc1c2bb8c5941a2a68d6daca7fd8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.788ex; height:2.509ex;" alt="{\displaystyle H'}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0d20eca58300d3e12e1a099bd96e616d32bd426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \vartriangleright }" loading="lazy"></td>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddcdb42f04e20193efceffe1e6bfefc2835bbf54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.241ex; height:2.509ex;" alt="{\displaystyle H''}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0d20eca58300d3e12e1a099bd96e616d32bd426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \vartriangleright }" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5acdcac635f883f8b4f0a01aa03b16b22f23b124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}" loading="lazy">

</td></tr></tbody></table>

<a href="Surjektiv" class="mw-redirect" title="Surjektiv">surjektiv</a> für jedes der Paare

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c64ca7167bae328e705cea185f12418a3f82939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.728ex; height:2.843ex;" alt="{\displaystyle (X,Y)\;\;\;\in }" loading="lazy"></td>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\Bigl \{}{\bigl (}G',H{\bigr )}{\Bigr .}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dd95285ccdf11cabdddcf875f4d62be74c49eb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.289ex; height:4.843ex;" alt="{\displaystyle {\Bigl \{}{\bigl (}G',H{\bigr )}{\Bigr .}}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8fa4cba3a446de313920e16251756e27312b825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle ,}" loading="lazy"></td>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bigl (}\varphi ^{-1}(H'),H'{\bigr )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcbb9137429b0e49759256b3fe725a7818d6971b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.402ex; height:3.343ex;" alt="{\displaystyle {\bigl (}\varphi ^{-1}(H'),H'{\bigr )}}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8fa4cba3a446de313920e16251756e27312b825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle ,}" loading="lazy"></td>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bigl (}\varphi ^{-1}(H''),H''{\bigr )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b7ce691f95bbbcc54784fbe66e1e6d8bdccdc18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.307ex; height:3.343ex;" alt="{\displaystyle {\bigl (}\varphi ^{-1}(H''),H''{\bigr )}}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8fa4cba3a446de313920e16251756e27312b825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle ,}" loading="lazy"></td>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\Bigl .}{\bigl (}G'',\{1\}{\bigr )}{\Bigr \}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d109a3c7bde563b5d92732c355dfc9221f83ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.812ex; height:4.843ex;" alt="{\displaystyle {\Bigl .}{\bigl (}G'',\{1\}{\bigr )}{\Bigr \}}.}" loading="lazy">

</td></tr></tbody></table>

Nun sind die Urbilder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{-1}(H')}">

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{-1}(H')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05bb9679df63557dd17ce412bfe31527544c1ba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.45ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}(H')}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{-1}(H'')}">

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{-1}(H'')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abdbf35244e79141c39480bfd0c870686ef8c841.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.903ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}(H'')}" loading="lazy"> Untergruppen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G',}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95451740ed592fc87d566234d7cef66d1799d0d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.158ex; height:2.843ex;" alt="{\displaystyle G',}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G''}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a37ec377e9bb29c7dd95a844c1b230fbbebea75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.964ex; height:2.509ex;" alt="{\displaystyle G''}" loading="lazy"> enthalten. Ferner ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\varphi ^{-1}(H'))=H'}">

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\varphi ^{-1}(H'))=H'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61cb135b4830322b5ec6da7fdad2684b0dd8b62c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.666ex; height:3.176ex;" alt="{\displaystyle \varphi (\varphi ^{-1}(H'))=H'}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\varphi ^{-1}(H''))=H'',}">

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\varphi ^{-1}(H''))=H'',}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7e46d889304343b7b2aaf39e84830b954a8c05e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.218ex; height:3.176ex;" alt="{\displaystyle \varphi (\varphi ^{-1}(H''))=H'',}" loading="lazy"> da alle <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"> ein Urbild in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76634fad5818a777669a77cd8c86d1d816e4c402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}" loading="lazy"> haben. Überdies ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{-1}(H'')}">

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{-1}(H'')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abdbf35244e79141c39480bfd0c870686ef8c841.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.903ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}(H'')}" loading="lazy"> ein Normalteiler von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{-1}(H').}">

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{-1}(H').}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4199464c9b3cdbfde6651064a788c8e11f957f0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.097ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}(H').}" loading="lazy">

<a href="Isomorphiesatz" title="Isomorphiesatz">Damit</a> ist der Subquotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H'/H''}">

<msup>

</msup>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e133f56a94114a159d33fc982427bbf9512830c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.191ex; height:3.009ex;" alt="{\displaystyle H'/H''}" 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"> als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H'/H''\cong \varphi ^{-1}(H')/\varphi ^{-1}(H'')}">

<msup>

</msup>

</mrow>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H'/H''\cong \varphi ^{-1}(H')/\varphi ^{-1}(H'')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c8fa3acf35371a9b16a8779d37f0e2f458109f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.805ex; height:3.176ex;" alt="{\displaystyle H'/H''\cong \varphi ^{-1}(H')/\varphi ^{-1}(H'')}" loading="lazy"> ein Subquotient 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"><a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Diskrete_Ordnung">Diskrete Ordnung</h3></div>

Die Ordnungsrelation »ist Subquotient von« ist bei <a href="Endliche_Gruppe" title="Endliche Gruppe">endlichen Gruppen</a> eine diskrete Ordnung, d. h. die von ihr erzeugte <a href="Ordnungstopologie" title="Ordnungstopologie">Ordnungstopologie</a> ist eine <a href="Diskrete_Topologie" title="Diskrete Topologie">diskrete Topologie</a>. In Formeln und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \preceq }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63dfe475e1377b3b4e936a3aa8fb1d7177dcdbc3.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 \preceq }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prec }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59707ac9078525b52a9e21a1baf9ab787af7a9aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \prec }" loading="lazy"> als Relationszeichen:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4896d24cd85a5a6e2a5473ba0a758d06e0ef02f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.744ex; height:2.509ex;" alt="{\displaystyle I\prec G,}" loading="lazy"> dann gibt es ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\prec G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8addd45a994e0f8338ff21ff1fd8df114aa2cc19.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\prec G}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\preceq H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1bba324bc4871f7dcff495ab74b5daa5aa131d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.334ex; height:2.343ex;" alt="{\displaystyle I\preceq H}" loading="lazy"> derart, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\preceq H'\prec G\implies H'=H.}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\preceq H'\prec G\implies H'=H.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba4462f659af0ea2f827ea176ee2e1e325549f00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.859ex; height:2.676ex;" alt="{\displaystyle H\preceq H'\prec G\implies H'=H.}" loading="lazy"></dd></dl>

Ein solches <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"> nennt man einen maximalen echten Subquotienten 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">. Der Begriff wird bspw. bei der Anordnung der sporadischen Gruppen im <a href="Hasse-Diagramm" title="Hasse-Diagramm">Hasse-Diagramm</a> benötigt.

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

/* 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/20130626081216/http://www.mathematik.uni-mainz.de/arbeitsgruppen/gruppentheorie/held/klassifi.pdf">Dieter Held: Die Klassifikation der endlichen einfachen Gruppen (PDF, 131 kB) S. 19</a> (<a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a> vom 26. Juni 2013 im <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>)

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Robert Griess: <cite style="font-style:italic">The Friendly Giant</cite>. In: <cite style="font-style:italic">Inventiones Mathematicae</cite>. Band 69, 1982, S. 91, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01389186">10.1007/BF01389186</a> (<a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PPN=PPN356556735_0069&DMDID=dmdlog7">Online bei digizeitschriften.de</a>).<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:Subquotient&rft.atitle=The+Friendly+Giant&rft.au=Robert+Griess&rft.btitle=Inventiones+Mathematicae&rft.date=1982&rft.doi=10.1007%2FBF01389186&rft.genre=book&rft.pages=91&rft.volume=69" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a rel="nofollow" class="external text" href="https://www.math.uni-bielefeld.de/~sek/top/leit/noether.htm">Die Noether'schen Isomorphie-Sätze</a>

</li>

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