Finite Groups with Two Class Sizes of Some Elements

Let G be a finite group. We prove that if {1,m} are the conjugacy class sizes of p-regular elements of primary and biprimary orders of G, for some prime p, then G has Abelian p-complement or G = PQ × A, with P ∈ Sylp (G), Q ∈ Sylq (G) and A ⊆ Z(G), with q a prime distinct from p. As a consequence, if {1, m} are the conjugacy class sizes of ab p-regular elements of primary and biprimary orders of G, then m = paqb. In particular, if b = 0 then G has abelian p-complement and if a = 0 then G = P × Q × A with A ⊆ Z(G)