Commuting Involutions in Finite Simple Groups

Abstract

We prove that if $G$ is a finite simple group and $x,y\in G$ are involutions, then $|x^G\cap C_G(y)|\to \infty$ as $|G|\to \infty$. This extends results of Guralnick-Robinson and Skresanov. We also prove a related result about $C_G(t)/O(C_G(t))$ that does not require the classification of finite simple groups.