About: Maschke's theorem

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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

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rdfs:label	Satz von Maschke (de) Teoremo de Maschke (eo) Teorema de Maschke (es) Théorème de Maschke (fr) Maschke's theorem (en) マシュケの定理 (ja) Теорема Машке (ru) 马施克定理 (zh) Теорема Машке (uk)
rdfs:comment	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 (es) 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. (fr) 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。 (ja) 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. (uk) 在代数中，马施克定理是有限群表示论中基本的定理之一。 (zh) Теорема Машке — теорема теории представлений, утверждающая при определённых условиях на характеристику поля, что всякое конечномерное представление конечной группы раскладывается в прямую сумму неприводимых. (ru) 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. (eo) 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. (de) 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 (en)
name	Corollary (en) Maschke's theorem (en) Maschke's Theorem (en)
dcterms:subject	Theorems in group theory Representation theory of finite groups Theorems in representation theory
Wikipage page ID	30876007 (xsd:integer)
Wikipage revision ID	1094452252 (xsd:integer)
Link from a Wikipage to another Wikipage	Multiplicity (mathematics) Semi-simplicity Jordan–Hölder theorem Unit (ring theory) Vector space Artin–Wedderburn theorem Mathematics Matrix (mathematics) Class function (algebra) Eigenvector Module theory Converse (logic) Corollary Theorems in group theory Heinrich Maschke Irreducible representation Field (mathematics) Graduate Texts in Mathematics Isomorphism Isomorphism of categories Mathematical proof Group (mathematics) Semisimple algebra Prime number Character theory Characteristic (algebra) Division ring Representation theory of finite groups Theorems in representation theory Splitting lemma Group representation Group ring Inner product Algebraically closed Order (group theory) Orthogonal complement Category of representations Rational number Real number Subrepresentation Finite group Schur orthogonality relations Semisimple module Simple module Springer-Verlag Direct sum of representations
Link from a Wikipage to an external page	https://archive.org/details/linearrepresenta1977serr
sameAs	Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem Maschke's theorem
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