Groups with conjugacy classes of coprime sizes

Abstract

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $⟨x^G⟩\cap ⟨y^G⟩$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $\ddot{I}€$-regular elements for some set of primes $\ddot{I}€$, then $x^G{y}^G$ is a $\ddot{I}€$-regular conjugacy class in $G$. The latter statement was previously known for $\ddot{I}€$-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 $p$.

13 pages. Fixed typo in arXiv abstract (article unchanged)