$$p$$ -Groups with few conjugacy classes of normalizers

Abstract

For a group \(G\), denote by \(\omega (G)\) the number of conjugacy classes of normalizers of subgroups of \(G\). Clearly, \(\omega (G)=1\) if and only if \(G\) is a Dedekind group. Hence if \(G\) is a 2-group, then \(G\) is nilpotent of class \(\le 2\) 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 \(\omega (G)\le p+1\). If \(p=2\), then \(G\) is of class \(\le 3\); if \(p>2\), then \(G\) is of class \(\le 2\).