Commutators between coprime order elements in non-abelian simple groups

Abstract

Recent investigations on the set of commutators between the elements of a finite group having relatively prime orders have prompt us to propose a variant of the Ore conjecture: For every finite non-abelian simple group and for every $g\in G$, there exist $x,y\in G$ with $g=[y,x]$ and with the order of $x$ relatively prime to the order of $y$. In this note we present some evidence towards the veracity of this conjecture by proving it for alternating groups and some sporadic simple groups.

Our main result was already proved by Pavel Shumyatsky in 2015 "Commutators of elements of coprime orders in finite groups" Forum Math. 27 (2015), 575 - 583 DOI 10.1515/ forum-2012-0127