Finite groups with some non-Abelian subgroups of non-prime-power order

Abstract

Let G be a finite group and NA(G) denote the number of conjugacy classes of all nonabelian subgroups of non-prime-power order of G. The Symbol π(G) denote the set of the prime divisors of |G|. In this paper we establish lower bounds on NA(G). In fact, we show that if G is a finite solvable group, then NA(G) = 0 or NA(G) ≥ 2|π(G)|−2, and if G is non-solvable, then NA(G) ≥ |π(G)| + 1. Both lower bounds are best possible.