University of IsfahanInternational Journal of Group Theory2251-76506320170901One-prime power hypothesis for conjugacy class sizes13191204310.22108/ijgt.2017.12043ENAlanCaminaUniversity of East AngliaRachelCaminaFitzwilliam College, University of CambridgeJournal Article20151102A 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.https://ijgt.ui.ac.ir/article_12043_81116a607ca09117e53c05589eda4c89.pdf