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About: Maschke's theorem
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Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. 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title="Switch to /sparql endpoint"><i class="bi-box-arrow-up-right"></i> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div> <!-- /navbar --> <!-- page-header --> <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6"><b>About:</b> <a href="http://dbpedia.org/resource/Maschke's_theorem">Maschke's theorem</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> <span class="text-nowrap">An Entity of Type: <a href="http://dbpedia.org/class/yago/Abstraction100002137">Abstraction100002137</a>, </span> <span class="text-nowrap">from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, </span> <span class="text-nowrap">within Data Space: <a href="http://dbpedia.org">dbpedia.org</a></span> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> <p class="lead">In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a represen</p> </div> </div> </div> </section> <!-- page-header --> <!-- property-table --> <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract"><small>dbo:</small>abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="de" >Der Satz von Maschke (nach Heinrich Maschke, 1899) ist eine zentrale Aussage aus dem mathematischen Teilgebiet der Darstellungstheorie endlicher Gruppen. Er besagt, dass Darstellungen außer im Spezialfall aus irreduziblen Darstellungen zusammengesetzt sind. Es seien eine endliche Gruppe und ein Körper. Das Wesen der Theorie der -linearen Darstellungen von hängt fundamental davon ab, ob die Charakteristik von ein Teiler der Ordnung von ist oder nicht. In ersterem Falle spricht man von modularen Darstellungen. Der Unterschied liegt im Wesentlichen in der Aussage des Satzes von Maschke begründet.</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="eo" >En matematiko, aŭ pli aparte en aparta grupa prezenta teorio, teoremo de Maschke estas la baza rezulto pruvanta, ke linearaj prezentoj de super kampoj de 0, kiel la kompleksaj, reelaj, kaj racionalaj nombroj, disfalas en neredukteblajn pecojn. Ĉi tio estas fundamenta, ekzemple, al la apliko de signaj tabeloj. Oni devas singardi, ĉar la prezento povas malkomponiĝi malsame super malsamaj kampoj: prezento povas esti nereduktebla super la reelaj nombroj sed ne super la kompleksaj nombroj. Pli ĝenerale, la teoremo veras por kampoj de pozitiva karakterizo p, kiel la , se la primo p ne dividas la ordon de G. Estu K esti kampo, G finia grupo, kaj estu KG la grupa algebro. Teoremo de Maschke statas ke kiel ringo, KG estas duone-simpla se kaj nur se la de K ne dividas la ordon de G. Sekve de tio de teoremo de Mascke, oni povas apliki la (iam nomatan kiel la struktura teoremo de Wedderburn) al KG. Kiam K estas la kompleksaj nombroj, ĉi tio montras, ke KG estas de kopioj de matrico (algebro), unu por ĉiu nereduktebla prezento.</span><small> (eo)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="es" >El teorema de Maschke, relativo a la teoría de representación de grupos, trata sobre la descomposición de la representación de un grupo finito en partes irreducibles. Si (V, ρ) es una representación de dimensión finita de un grupo finito sobre un cuerpo de característica cero, y U es un subespacio invariante de V, entonces el teorema afirma que U admite un complemento directo invariante W; es decir, la representación (V, ρ) es completamente reducible. El teorema puede ser generalizado para cuerpos de característica finita. * Datos: Q656198</span><small> (es)</small></span></li> <li><span class="literal"><span property="dbo:abstract" lang="en" >In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="fr" >En mathématiques et plus précisément en algèbre, le théorème de Maschke est un des théorèmes fondamentaux de la théorie des représentations d'un groupe fini. Ce théorème établit que si la caractéristique du corps ne divise pas l'ordre du groupe, alors toute représentation se décompose en facteurs irréductibles. Il se reformule en termes de modules sur l'algèbre d'un groupe fini et possède une généralisation partielle aux groupes compacts. Ce théorème doit son nom au mathématicien allemand Heinrich Maschke.</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ja" >数学、特に群の表現論においてマシュケの定理（マシュケのていり、英: Maschke's theorem）とは、有限群の表現の既約表現への分解に関する定理である。に名を因む。有限群 G のある標数 0 の体上の有限次元表現 (V, ρ) に対し、任意の G-不変部分空間 U は G-不変な直和補因子 W を持つこと、言い換えれば、表現 (V, ρ) がであることを述べるものである。より一般に、有限体のような正標数 p の体に対しても、p が群 G の位数を割り切らないならば、マシュケの定理は成り立つ。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="uk" >В математиці, теорема Машке, — теорема в теорії представлень груп щодо розкладу представлень скінченних груп на . Теорема Машке дозволяє робити висновки про представленя скінченних груп G без їх обчислень. Вона зводить задачу класифікації всіх представлень до задачі класифікації незвідних представлень, на пряму суму яких розкладається довільне представлення.</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="zh" >在代数中，马施克定理是有限群表示论中基本的定理之一。</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ru" >Теорема Машке — теорема теории представлений, утверждающая при определённых условиях на характеристику поля, что всякое конечномерное представление конечной группы раскладывается в прямую сумму неприводимых.</span><small> (ru)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageExternalLink"><small>dbo:</small>wikiPageExternalLink</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="dbo:wikiPageExternalLink nofollow" 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text-break"><ul> <li><span class="literal"><span property="dbp:mathStatement" lang="en" >Every representation of a finite group over a field with characteristic not dividing the order of is a direct sum of irreducible representations.</span><small> (en)</small></span></li> <li><span class="literal"><span property="dbp:mathStatement" lang="en" >Let be a finite group and a field whose characteristic does not divide the order of . Then , the group algebra of , is semisimple.</span><small> (en)</small></span></li> <li><span class="literal"><span property="dbp:mathStatement" lang="en" >If is a group and is a field with characteristic not dividing the order of , then the category of representations of over is semi-simple.</span><small> (en)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/name"><small>dbp:</small>name</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><span property="dbp:name" lang="en" >Corollary</span><small> (en)</small></span></li> <li><span class="literal"><span property="dbp:name" lang="en" >Maschke's theorem</span><small> (en)</small></span></li> <li><span class="literal"><span property="dbp:name" lang="en" >Maschke's Theorem</span><small> (en)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" 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class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Message106598915" href="http://dbpedia.org/class/yago/Message106598915"><small>yago</small>:Message106598915</a></span></li> <li><span class="literal"><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Proposition106750804" href="http://dbpedia.org/class/yago/Proposition106750804"><small>yago</small>:Proposition106750804</a></span></li> <li><span class="literal"><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Statement106722453" href="http://dbpedia.org/class/yago/Statement106722453"><small>yago</small>:Statement106722453</a></span></li> <li><span class="literal"><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Theorem106752293" href="http://dbpedia.org/class/yago/Theorem106752293"><small>yago</small>:Theorem106752293</a></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment"><small>rdfs:</small>comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="es" >El teorema de Maschke, relativo a la teoría de representación de grupos, trata sobre la descomposición de la representación de un grupo finito en partes irreducibles. Si (V, ρ) es una representación de dimensión finita de un grupo finito sobre un cuerpo de característica cero, y U es un subespacio invariante de V, entonces el teorema afirma que U admite un complemento directo invariante W; es decir, la representación (V, ρ) es completamente reducible. El teorema puede ser generalizado para cuerpos de característica finita. * Datos: Q656198</span><small> (es)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="fr" >En mathématiques et plus précisément en algèbre, le théorème de Maschke est un des théorèmes fondamentaux de la théorie des représentations d'un groupe fini. Ce théorème établit que si la caractéristique du corps ne divise pas l'ordre du groupe, alors toute représentation se décompose en facteurs irréductibles. Il se reformule en termes de modules sur l'algèbre d'un groupe fini et possède une généralisation partielle aux groupes compacts. Ce théorème doit son nom au mathématicien allemand Heinrich Maschke.</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ja" >数学、特に群の表現論においてマシュケの定理（マシュケのていり、英: Maschke's theorem）とは、有限群の表現の既約表現への分解に関する定理である。に名を因む。有限群 G のある標数 0 の体上の有限次元表現 (V, ρ) に対し、任意の G-不変部分空間 U は G-不変な直和補因子 W を持つこと、言い換えれば、表現 (V, ρ) がであることを述べるものである。より一般に、有限体のような正標数 p の体に対しても、p が群 G の位数を割り切らないならば、マシュケの定理は成り立つ。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="uk" >В математиці, теорема Машке, — теорема в теорії представлень груп щодо розкладу представлень скінченних груп на . Теорема Машке дозволяє робити висновки про представленя скінченних груп G без їх обчислень. Вона зводить задачу класифікації всіх представлень до задачі класифікації незвідних представлень, на пряму суму яких розкладається довільне представлення.</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="zh" >在代数中，马施克定理是有限群表示论中基本的定理之一。</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ru" >Теорема Машке — теорема теории представлений, утверждающая при определённых условиях на характеристику поля, что всякое конечномерное представление конечной группы раскладывается в прямую сумму неприводимых.</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="eo" >En matematiko, aŭ pli aparte en aparta grupa prezenta teorio, teoremo de Maschke estas la baza rezulto pruvanta, ke linearaj prezentoj de super kampoj de 0, kiel la kompleksaj, reelaj, kaj racionalaj nombroj, disfalas en neredukteblajn pecojn. Ĉi tio estas fundamenta, ekzemple, al la apliko de signaj tabeloj. Oni devas singardi, ĉar la prezento povas malkomponiĝi malsame super malsamaj kampoj: prezento povas esti nereduktebla super la reelaj nombroj sed ne super la kompleksaj nombroj.</span><small> (eo)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="de" >Der Satz von Maschke (nach Heinrich Maschke, 1899) ist eine zentrale Aussage aus dem mathematischen Teilgebiet der Darstellungstheorie endlicher Gruppen. Er besagt, dass Darstellungen außer im Spezialfall aus irreduziblen Darstellungen zusammengesetzt sind.</span><small> (de)</small></span></li> <li><span class="literal"><span property="rdfs:comment" lang="en" >In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a represen</span><small> (en)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label"><small>rdfs:</small>label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="de" >Satz von Maschke</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="eo" >Teoremo de Maschke</span><small> (eo)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="es" >Teorema de Maschke</span><small> (es)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="fr" >Théorème de Maschke</span><small> (fr)</small></span></li> <li><span class="literal"><span property="rdfs:label" lang="en" >Maschke's theorem</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ja" >マシュケの定理</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ru" >Теорема Машке</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="zh" >马施克定理</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="uk" >Теорема Машке</span><small> (uk)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs"><small>owl:</small>sameAs</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.02ryvg" href="http://rdf.freebase.com/ns/m.02ryvg"><small>freebase</small>:Maschke's theorem</a></span></li> <li><span class="literal"><a class="uri" rel="owl:sameAs" 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