Groups with few self-centralizing subgroups which are not self-normalizing

Abstract

A self-normalizing subgroup is always self-centralizing, but the converse is not necessarily true. Given a finite group $G$, we denote by $w(G)$ the number of all self-centralizing subgroups of $G$ which are not self-normalizing. We observe that $w(G)=0$ if and only if $G$ is abelian, and that if $G$ is nonabelian nilpotent then $w(G)\ge 3$. We also prove that if $w(G)\le 20$ then $G$ is solvable. Finally, we provide structural information in the case when $w(G)\le 3$.