On A-groups with the same index set as a nilpotent group

Abstract

Let G be a finite group and $N\left(G\right)$ be the set of conjugacy class sizes of G. For a prime p, let $\left|G\right|{|}_p$ be the highest p-power dividing some element of $N\left(G\right)$ and define $\left|G\right||={\Pi }_{p\in \pi \left(G\right)}|G|{|}_p$. 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 $N\left(G\right)$ contains $\left|G\right|{|}_p$ for every $p\in \pi \left(G\right)$ as well as $\left|G\right||$, then G must be abelian. This result gives a positive answer to a question posed by Camina and Camina in 2006.