Finite groups with few conjugacy classes of non-cyclic subgroups

Abstract

Let G be a finite group and δ(G) denote the number of conjugacy classes of all non-cyclic subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, a group G is called δπ-group if δ(G) ≤ |π(G)| + 1. We show all δπ-groups are solvable. Furthermore, δπ-groups are classi ed completely and non-solvable groups with δ(G) = |π(G)| + 2 must be isomorphic to A₅ or SL(2, 5).