Groups with conjugacy classes of coprime sizes

TL;DR

This paper investigates the structure of finite groups with conjugacy classes of coprime sizes, proving that certain generated subgroups are abelian and extending known results on $ ext{pi}$-regular elements to more general groups.

Contribution

It generalizes previous results on $ ext{pi}$-regular elements from $ ext{pi}$-separable groups to all finite groups, revealing new structural properties.

Findings

The intersection of generated subgroups from conjugacy classes of coprime sizes is abelian and normal.

If elements are $ ext{pi}$-regular with coprime class sizes, their product forms a $ ext{pi}$-regular conjugacy class.

Extension of results on the common divisor graph to broader classes of finite groups.

Abstract

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $\pi$-regular elements for some set of primes $\pi$, then $x^G y^G$ is a $\pi$-regular conjugacy class in $G$. The latter statement was previously known for $\pi$-separable groups $G$ and this generalisation permits us to extend several results concerning the common divisor graph on $p$-regular conjugacy classes, for some prime $p$.