Products of groups and class sizes of $\pi$-elements

Abstract

We provide structural criteria for some finite factorised groups $G=AB$ when the conjugacy class sizes in $G$ of certain $\pi$-elements in $A\cup B$ are either $\pi$-numbers or ${\pi }^{\prime }$-numbers, for a set of primes $\pi$. In particular, we extend for products of groups some earlier results.