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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Subnormalteiler</span></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"><p>In der <a href="Gruppentheorie" title="Gruppentheorie">Gruppentheorie</a> wird eine <a href="Untergruppe" title="Untergruppe">Untergruppe</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> einer <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>G</mi>
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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
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</math></span><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"></span> als <b>Subnormalteiler</b> (oder <b>subnormale Untergruppe</b>) bezeichnet, falls eine <a href="Reihe_(Gruppentheorie)" title="Reihe (Gruppentheorie)">Subnormalreihe</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
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</math></span><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"></span> nach <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> existiert, das heißt, falls es eine endliche Kette <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=S_{0}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
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<mi>S</mi>
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<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle S=S_{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/baa07dbad9716a470adce6814bffc39ffe00eb4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.077ex; height:2.509ex;" alt="{\displaystyle S=S_{0}}" loading="lazy"></span>&nbsp;≤&nbsp;…&nbsp;≤&nbsp;<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{k}=G}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
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<mi>k</mi>
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<mo>=</mo>
<mi>G</mi>
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<annotation encoding="application/x-tex">{\displaystyle S_{k}=G}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95c6428847daea463043bcc86d207b37dfbbed55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.439ex; height:2.509ex;" alt="{\displaystyle S_{k}=G}" loading="lazy"></span> von Untergruppen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
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</math></span><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"></span> gibt, so dass jeweils <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
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<annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de6e810a93f67802ecb603ee0e3324005c6e583e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}" loading="lazy"></span> <a href="Normalteiler" title="Normalteiler">Normalteiler</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i+1}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
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<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle S_{i+1}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a289234ec56eb141018905b8883e31cfbe7fd7ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.325ex; height:2.509ex;" alt="{\displaystyle S_{i+1}}" loading="lazy"></span> ist.
</p><p>Subnormalteiler wurden – noch unter der Bezeichnung <i>nachinvariante Untergruppe</i> – erstmals von <a href="Helmut_Wielandt" title="Helmut Wielandt">Helmut Wielandt</a> in seiner 1939 erschienenen Habilitationsschrift <i>Eine Verallgemeinerung der invarianten Untergruppen</i> betrachtet. Wielandt konnte unter anderem zeigen, dass in <a href="Endliche_Gruppe" title="Endliche Gruppe">endlichen Gruppen</a> das Erzeugnis zweier Subnormalteiler stets wieder subnormal ist, die Subnormalteiler also einen <a href="Verband_(Mathematik)" title="Verband (Mathematik)">Verband</a> bilden.
</p><p>Der Begriff des Subnormalteilers ist insofern eine Verallgemeinerung des Begriffs des Normalteilers, als ein Subnormalteiler nicht unbedingt normal in der ganzen Gruppe sein muss. Jeder Normalteiler ist aber stets ein Subnormalteiler.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Die von einer Spiegelung erzeugte Untergruppe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\{e,(12)(34)\}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>e</mi>
<mo>,</mo>
<mo stretchy="false">(</mo>
<mn>12</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>34</mn>
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<annotation encoding="application/x-tex">{\displaystyle Z=\{e,(12)(34)\}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dce2641c2600f94757ddf04418aaee04d0758489.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.49ex; height:2.843ex;" alt="{\displaystyle Z=\{e,(12)(34)\}}" loading="lazy"></span> der <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrischen Gruppe</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{4}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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<annotation encoding="application/x-tex">{\displaystyle S_{4}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc1632bc1d95d33ccb5473b9d8cc333c2dd0d13a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{4}}" loading="lazy"></span> ist ein Normalteiler der <a href="Kleinsche_Vierergruppe" title="Kleinsche Vierergruppe">Kleinschen Vierergruppe</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
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</math></span><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"></span>, welche wiederum normal in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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</msub>
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<annotation encoding="application/x-tex">{\displaystyle S_{4}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc1632bc1d95d33ccb5473b9d8cc333c2dd0d13a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{4}}" loading="lazy"></span> liegt. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> ist also Subnormalteiler von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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<annotation encoding="application/x-tex">{\displaystyle S_{4}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc1632bc1d95d33ccb5473b9d8cc333c2dd0d13a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{4}}" loading="lazy"></span>, allerdings kein Normalteiler, da <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((12)(34))^{(123)}=(13)(24)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mo stretchy="false">(</mo>
<mn>12</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>34</mn>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mn>123</mn>
<mo stretchy="false">)</mo>
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</msup>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>13</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>24</mn>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle ((12)(34))^{(123)}=(13)(24)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e68e64ceedf3bae8572f6861349462bb1ada52e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.422ex; height:3.343ex;" alt="{\displaystyle ((12)(34))^{(123)}=(13)(24)}" loading="lazy"></span> nicht in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> liegt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Helmut Wielandt: <i>Eine Verallgemeinerung der invarianten Untergruppen.</i> In: Mathematische Zeitschrift <b>45</b> (1939), S. 209–244.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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