One-prime power hypothesis for conjugacy class sizes

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_۲(q) for q in {۴,۸}. We confirm this conjecture.

Keywords:

conjugacy class sizes , Finite groups , insoluble groups , bipartite graph