Maschke's 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 doesn't divide the order of G.

Read more about Maschke's TheoremReformulation and The Meaning

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Maschke's Theorem - Reformulation and The Meaning
... 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 ... well developed theory of semisimple rings, in particular, the Artin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem) ... Returning to representation theory, Maschke's theorem and its module-theoretic version allow one to make general conclusions about representations of a ...
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... This is a consequence of Maschke's theorem, which states that any subrepresentation V of a G-representation W has a G-invariant complement ... Maschke's theorem holds more generally for fields of positive characteristic p, such as the finite fields, as long as the prime p is coprime to the ... Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation ...

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