On the number of conjugacy classes of π-elements in finite groups

Let G be a finite group and pi be a set of primes. Put , where is the number of conjugacy classes of pi-elements in G and |G| (pi) is the pi-part of the order of G. In this paper we initiate the study of this invariant by showing that if then G possesses an abelian Hall pi-subgroup, all Hall pi-subgroups of G are conjugate, and every pi-subgroup of G lies in some Hall pi-subgroup of G. Furthermore, we have or . This extends and generalizes a result of W. H. Gustafson.