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

Hassanzadeh, Mahmoud; Mostaghim, Zohreh

Rendiconti del Seminario Matematico della Università di Padova, Tome 142 (2019), pp. 69-80.

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

Classification : 20, 00
Mots-clés : Self-centralizing subgroup, self-normalizing subgroup, minimal simple group, A-group