Conjugacy classes of non-normal subgroups in finite nilpotent groups

摘要

Let nu(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. Poland and Rhemtulla [10] proved that if G is nilpotent of class c then nu(G) >= c - 1 unless G is a Hamiltonian group. The sharpness of this lower bound is a problem about finite p-groups, since nu(H x K) >= nu(H)nu(K) + nu(H)mu(K) + mu(H)nu(K), where mu(G) denotes the number of normal subgroups of G. In this paper, we show that the bound nu(G) >= c - 1 can be substantially improved: if G is a finite p-group and vertical bar G'vertical bar = p(k), then nu(G) >= 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 nu(G) >= 2(k - 1).