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

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=PSL(n,2) for n≥4. More precisely, we show that the conjecture does not hold for the alternating group A8n for all n≥2.