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^Gy^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.