Commutators of elements of coprime orders in finite groups

TL;DR

This paper investigates properties of finite groups related to commutators of elements with coprime orders, providing a solubility criterion and confirming a conjecture for alternating groups.

Contribution

It introduces a new criterion for group solubility based on coprime order commutators and confirms a conjecture for alternating groups.

Findings

A criterion for the solubility of finite groups using coprime order commutators

Confirmation of the conjecture for alternating groups

Proposal that every element in a nonabelian simple group is a coprime order commutator

Abstract

This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture that every element of a nonabelian simple group is a commutator of elements of coprime orders and we confirm this conjecture for the alternating groups.