On A-Groups with the Same Index Set as a Nilpotent Group

Let G be a finite group and $N(G)$ be the set of conjugacy class sizes of G. For a prime p, let |G||ₚ be the highest p-power dividing some element of $N(G)$. and define |G|| = Πp∈ π(G)|G||ₚ. G is said to be an A-group if all its Sylow subgroups are abelian. We prove that if G is an A-group such that $N(G)$ contains |G||ₚ for every p∈ π(G) as well as $|G||$, then G must be abelian. This result gives a positive answer to a question posed by Camina and Camina in 2006.