Orders of commutators and Products of conjugacy classes in finite groups

Abstract

Let $G$ be a finite group, let $x \in G$, and let $p$ be a prime. We prove that the commutator $[x,g]$ is a $p$-element for every $g \in G$ if and only if $x$ is central modulo $\mathbf{O}_p(G)$, where $\mathbf{O}_p(G)$ denotes the largest normal $p$-subgroup of $G$. This result provides a common generalization of certain variants of both the Baer--Suzuki theorem and Glauberman's $\mathbf{Z}_p^*$-theorem. As an application, we show that if $K$ is a conjugacy class of $G$ such that $K^{-1}K = 1 \cup D \cup D^{-1}$ for some conjugacy class $D$ of $G$, then the subgroup generated by $K$ is solvable.