Groups with conjugacy classes of coprime sizes

Suppose that x, y are elements of a finite group G lying in conjugacy classes of coprime sizes. We prove that ⟨x^G⟩ ∩ ⟨y^G⟩ is an abelian normal subgroup of G and, as a consequence, that if x and y are π‐regular elements for some set of primes π, then x^G y^G is a π‐regular conjugacy class in G. The latter statement was previously known for π‐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.