Determining hypercentral Hall subgroups in finite groups

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