Counterexamples to a conjecture of M. Pellegrini and P. Shumyatsky

TL;DR

This paper presents counterexamples to a conjecture about the distribution of odd order elements in cosets of centralizers of involutions in finite simple groups, specifically disproving it for certain alternating groups.

Contribution

The authors provide explicit counterexamples to a conjecture regarding involution centralizers in finite simple groups, expanding understanding of their structure.

Findings

Counterexamples found in alternating groups $A_{8n}$ for all $n extgreater{}2$

The conjecture does not hold universally outside the case $G=PSL(n,2)$ for $n extgreater{}3$

The result refutes a previously believed property of involution centralizers in finite simple groups.

Abstract

In this article, we provide counterexamples to a conjecture of M. Pellegrini and P. Shumyatsky which states that each coset of the centralizer of an involution in a finite non-abelian simple group $G$ contains an odd order element, unless $G=\text{PSL}(n,2)$ for $n\geq 4$. More precisely, we show that the conjecture does not hold for the alternating group $A_{8n}$ for all $n\geq 2$.