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

摘要

Let G be a finite group and π be a set of primes. Put d_π(G) =k_π(G)/|G|_π, where k_π(G) is the number of conjugacy classes of π-elements in G and |G|_π is the π-part of the order of G. In this paper we initiate the study of this invariant by showing that if d_π(G) > 5/8 then G possesses an abelian Hall π-subgroup, all Hall π-subgroups of G are conjugate, and every π-subgroup of G lies in some Hall π-subgroup of G. Furthermore, we have d_π(G) = 1 or d_π(G) = 2/3. This extends and generalizes a result of W. H. Gustafson.