Maschke's Theorem

A criterion in representation theory stating that for a finite group G and a field k whose characteristic does not divide |G|, the group algebra k[G] is semisimple; equivalently every finite-dimensional k-representation of G decomposes as a direct sum of irreducible representations.

A criterion for complete reducibility of representations of a finite group G over a field k: the group algebra kG is semisimple (equivalently every finite-dimensional representation is completely reducible) precisely when the characteristic of k does not divide the order of G.