Maschke theorem

In mathematics, in particular group representation theory, Maschke's theorem is the basic result proving that linear representations of a finite group over the complex numbers break up into irreducible pieces. This is fundamental, for example, to the application of character tables. As a piece of abstract algebra, the theorem characterises the good cases, for which the required behaviour will hold, in terms of the properties of the underlying field of scalars. For example, the same type of result holds for the real number field, though which representations are irreducible changes.

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

It is related to the Artin-Wedderburn theorem, which gives a decomposition of a finite-dimensional semisimple algebra into a finite product of algebras of matrices. In fact Maschke's theorem is a justification for applying the Artin-Wedderburn theorem to the group algebra.