Gustavo A. Fernández-Alcober, Leire Legarreta
May 21, 2008
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. Algebra 24 (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).