Maschke theorem

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. If (V, ρ) is a finite-dimensional representation of a finite group G over a field of characteristic zero, and U is an invariant subspace of V, then the theorem claims that U admits an invariant direct complement W; in other words, the representation (V, ρ) is completely reducible. More generally, the theorem holds for fields of positive characteristic p, such as the finite fields, if the prime p does not divide the order of G.

One of the approaches to representations of finite groups is through module theory. Representations of a group G are replaced by modules over its group algebra K[G]. Irreducible representations correspond to simple modules. Maschke's theorem addresses the question: is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? In the module-theoretic language, is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

Maschke's Theorem. Let G be a finite group and K a field whose characteristic does not divide the order of G. Then K[G], the group algebra of G, is semisimple.

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