Itô's theorem on groups with two class sizes revisited

Abstract

Let G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p'-elements of prime power order, say 1 and m, then m = p(a)q(b), for two distinct primes p and q, and G either has an abelian p-complement or G = PQ x A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Ito's theorem on groups having exactly two class sizes for elements of prime power order.