Maschke theorem

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In mathematics, or more specifically in particular group representation theory, Maschke's theorem states:

Let K be a field, G a finite group, and let KG denote the group algebra. Then as a ring, KG is semi-simple if and only if the characteristic of K does not divide the order of G.

Maschke's theorem is the basic result proving that linear representations of a finite group over fields of characteristic 0, such as the complex, real, and rational numbers, break up into irreducible pieces. This is fundamental, for example, to the application of character theory.

One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real numbers but not over the complex numbers.

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.

As a consequence of Maschke's theorem, we can apply the Artin-Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem) to KG. When K is the complex numbers, this shows that KG is a direct sum of copies of matrix algebras, one for each irreducible representation.