Conjugacy classes of non-normal subgroups in finite nilpotent groups

Abstract

Let v(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. Poland and Rhemtulla [J. Poland and A. Rhemtulla. The number of conjugacy classes of non-normal subgroups in nilpotent groups. Comm. Algebra24 (1996), 3237–3245.] proved that if G is nilpotent of class c then c − 1 unless G is a Hamiltonian group. The sharpness of this lower bound is a problem about finite p-groups, since , where μ(G) denotes the number of normal subgroups of G. In this paper, we show that the bound c − 1 can be substantially improved: if G is a finite p-group and , then p(k − 1) + 1, unless G is a Hamiltonian group or a generalized quaternion group. In these exceptional cases, G is a 2-group and 2(k − 1).