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 $⟨x^G⟩\cap ⟨y^G⟩$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $\pi$-regular elements for some set of primes $\pi$, then $x^G{y}^G$ is a $\pi$-regular conjugacy class in $G$. The latter statement was previously known for $\pi$-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 $p$.