Maschke’s theorem

E924213

mathematical theorem theorem theorem in representation theory

Maschke’s theorem is a fundamental result in representation theory stating that every finite group representation over a field of characteristic not dividing the group order is completely reducible into a direct sum of irreducible representations.

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Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem ⓘ theorem in representation theory ⓘ
appliesTo	finite groups ⓘ group representations ⓘ linear representations of finite groups ⓘ
assumes	characteristic of the field does not divide the group order ⓘ group order is finite ⓘ
concerns	direct sum decompositions ⓘ finite-dimensional modules ⓘ irreducible representations ⓘ modules over group algebras ⓘ
concludes	every finite-dimensional representation is completely reducible ⓘ every representation decomposes as a direct sum of irreducible representations ⓘ every subrepresentation has a complementary subrepresentation ⓘ
equivalentTo	semisimplicity of the group algebra over the field ⓘ
failsWhen	field characteristic divides the group order ⓘ
field	representation theory ⓘ
hasConsequence	characters of irreducible representations form an orthonormal basis of class functions ⓘ every finite group representation over C is completely reducible ⓘ regular representation decomposes into a direct sum of irreducibles ⓘ
hasProofTechnique	averaging over the group ⓘ construction of invariant complements ⓘ
holdsOver	fields of characteristic p not dividing the group order ⓘ fields of characteristic zero ⓘ
implies	category of finite-dimensional representations is semisimple ⓘ group algebra is semisimple ⓘ
isTaughtIn	graduate algebra courses ⓘ representation theory courses ⓘ
namedAfter	Heinrich Maschke NERFINISHED ⓘ
originalLanguage	German NERFINISHED ⓘ
originallyPublishedIn	Mathematische Annalen NERFINISHED ⓘ
relatedTo	Schur’s lemma NERFINISHED ⓘ Wedderburn’s theorem NERFINISHED ⓘ complete reducibility ⓘ group algebra ⓘ semisimple module ⓘ
requires	field whose characteristic does not divide the group order ⓘ finite group ⓘ
subfield	group representation theory ⓘ
usedIn	Fourier analysis on finite groups NERFINISHED ⓘ character theory of finite groups ⓘ classification of irreducible representations of finite groups ⓘ construction of projection operators onto irreducible components ⓘ decomposition of group representations ⓘ harmonic analysis on finite groups ⓘ
yearProved	1899 ⓘ

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Methods of Representation Theory → covers → Maschke’s theorem ⓘ