TeX -Groups with few conjugacy classes of normalizers

Abstract

For a group G , denote by ω(G) the number of conjugacy classes of normalizers of subgroups of G . Clearly, ω(G)=1 if and only if G is a Dedekind group. Hence if G is a 2-group, then G is nilpotent of class ≤2 and if G is a p -group, p>2 , then G is abelian. We prove a generalization of this. Let G be a finite p -group with ω(G)≤p+1 . If p=2 , then G is of class ≤3 ; if p>2 , then G is of class ≤2 .