 The celebrated Feit–Thompson theorem states that every finite group of odd order is solvable.In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.The group "S"5 is not solvable — it has a composition series {E, "A"5, "S"5} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to "A"5 and "C"2; and "A"5 is not abelian.