One-prime power hypothesis for conjugacy class sizes

Abstract

A finite group $G$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to $S times A$ where $A$ is abelian and $S cong PSL_2(q)$ for $q in {4,8}$. We confirm this conjecture.