Corollary to Maschke's Theorem.

Question

If in Maschke's Theorem, for group ring KG where K is any field s.t. char $K \nmid |G|$, I take G to be finite , then I know Maschke implies that KG will be semisimple so it is isomorphic to a direct sum of Matrices over division rings, but how does adding that G is also abelian implies that KG will be direct sum of fields?