Leire Legarreta

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. 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).

Some restrictions on normalizers or centralizers in finite p-groups

We study three restrictions on normalizers or centralizers in finite p-groups, namely: (i) |NG(H):H|≤pk for every H

Bounds for the Number of Conjugacy Classes of Non-Normal Subgroups in a Finite p-Group

\triangleleft / -

Some finiteness conditions on normalizers or centralizers in groups

G, (ii) |NG(〈g〉):〈g〉|≤pk for every 〈g〉

A restriction on centralizers in finite groups

\triangleleft / -

Non-inner automorphisms of order p in finite normally constrained p-groups

G, and (iii) |CG(g): 〈g〉 ≤ pk for every 〈g〉