Commuting Involutions in Finite Simple Groups

TL;DR

This paper investigates the behavior of involutions in finite simple groups, showing that the intersection of conjugacy classes and centralizers grows unboundedly with the group's size, extending previous results and providing new insights without relying on classification.

Contribution

It proves that the intersection size of involution conjugacy classes and centralizers tends to infinity in large finite simple groups, extending prior work and offering classification-independent results.

Findings

Intersection size of involution conjugacy classes grows with group size

Results extend previous work of Guralnick-Robinson and Skresanov

Provides classification-free analysis of centralizer quotients

Abstract

We prove that if $G$ is a finite simple group and $x, y \in G$ are involutions, then $|x^G \cap C_G(y)| \rightarrow \infty$ as $|G| \rightarrow \infty$. 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.