Finite groups with three conjugacy class sizes of certain elements

Let G be a finite group and m, n two positive coprime integers. We prove that the set of conjugacy class sizes of primary and biprimary elements of G is {1, m, n} if and only if G is quasi-Frobenius with abelian kernel and complement. Throughout this paper all groups considered are finite and G always denotes a group. A primary element is an element of prime power order and a biprimary element is an element whose order is divisible by precisely two distinct primes. We will denote by x G the conjugacy class containing x, and |x G | the conjugacy class size of x G. A positive integer a is a Hall number of group G if a is a divisor of |G| and (a, |G|/a) = 1. We say that G is quasi-Frobenius if G/Z(G) is Frobenius. The inverse image in G of the kernel and a complement of G/Z(G) are called the kernel and a complement of G. The other notation and terminology are standard, as in (2).