On an extension of a theorem on conjugacy class sizes

Abstract

Let G be a finite group. We extend Alan Camina’s theorem on conjugacy classes sizes which asserts that if the conjugacy classes sizes of G are {1, p ^a, q ^b, p ^a q ^b}, where p and q are two distinct primes and a and b are integers, then G is nilpotent. We show that let G be a group and assume that the conjugacy classes sizes of elements of primary and biprimary orders of G are exactly {1, p ^a, n,p ^a n} with (p, n) = 1, where p is a prime and a and n are positive integers. If there is a p-element in G whose index is precisely p ^a, then G is nilpotent and n = q ^b for some prime q ≠ p.