Determining Hypercentral Hall Subgroups in Finite Groups

Let G be a finite group, and let \(\pi \) $\pi$ be a set of primes. The aim of this paper is to obtain some results concerning how much information about the \(\pi \) $\pi$ -structure of G can be gathered from the knowledge of the sizes of conjugacy classes of its \(\pi \) $\pi$ -elements and of their multiplicities. Among other results, we prove that this multiset of class sizes determines whether G has a hypercentral Hall \(\pi \) $\pi$ -subgroup.