Coprime commutators in PSL(2,q)

TL;DR

This paper proves that every element in PSL(2,q) can be expressed as a commutator of elements with coprime orders, using conjugation properties of involutions.

Contribution

It establishes that all elements of PSL(2,q) are coprime commutators, revealing a new structural property of this group.

Findings

Every element of PSL(2,q) is a coprime commutator.

Any two involutions in PSL(2,q) are conjugate by an element of odd order.

The proof relies on conjugation properties of involutions in PSL(2,q).

Abstract

We show that every element of PSL(2,q) is a commutator of elements of coprime orders. This is proved by showing first that in PSL(2,q) any two involutions are conjugate by an element of odd order.