Groups with conjugacy classes of coprime sizes

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 .