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 A 2 n for all n ≥ 4.