ItÔ's Theorem on Groups with two class sizes revisited

LetG be a finite p-solvable group. We prove that ifG has exactly two conjugacy class sizes of p 0 -elements of prime power order, say 1 and m, then m = p aq b , for two distinct primes p and q, and G either has an abelian p-complement or G = PQ 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 Itô’s theorem on groups having exactly two class sizes for elements of prime power order.This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the second and third authors are supported by the Valencian Government, Proyecto PROMETEO/2011/30. The second author is also supported by grant Fundacio Caixa-Castell ´ o P11B2010-47