Groups with conjugacy classes of coprime sizes
Rachel D. Camina, Attila Mar\'oti, Emanuele Pacifici, Chris Parker, Kamilla Rekv\'enyi, Jack Saunders, V\'ictor Sotomayor, Gareth Tracey, Martin van Beek

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