A Huppert-type $\rho^*$--$\sigma^*$ theorem for sub-class sizes

Abstract

For a finite group $G$, let $\rho^{*}(G)$ be the set of primes dividing some sub-class size of $G$, and let $\sigma^{*}(G)$ be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's $\rho$--$\sigma$ problem, we prove that \[ |\rho^{*}(G)|<2\sigma^{*}(G) \] for every finite nonnilpotent group $G$. This answers \cite[Remark~5.3]{QY} and extends to arbitrary finite nonnilpotent groups the coefficient-$2$ bound proved in \cite[Theorem~1.7]{QY} for solvable nonnilpotent groups. The coefficient $2$ is best possible. The proof does not use the classification of finite simple groups.