A note on a theorem on conjugacy class sizes

Abstract

Let \(p\) be a fixed prime, \(a\) and \(n\) are positive integers such that \((p,n) = 1\). It is shown that if \(G\) is a finite group such that for every prime \(q\) the set of the conjugacy class sizes of all \(\{ p,q\} -\) elements of \(G\) is \(\left\{ {1,p^{a} {,}n,p^{a} n} \right\}\), and there is a \(p -\) element in \(G\) whose conjugacy class has size \(p^{a}\), then \(G\) is nilpotent and \(n\) is a prime power.