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

### Mahmoud Hassanzadeh

Iran University of Science & Technology, Tehran, Iran### Zohreh Mostaghim

Iran University of Science & Technology, Tehran, Iran

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## 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 $w(G)\geq 3$. We also prove that if $w(G)\leq 20$ then $G$ is solvable. Finally, we provide structural information in the case when $w(G)\leq 3$.

## Cite this article

Mahmoud Hassanzadeh, Zohreh Mostaghim, Groups with few self-centralizing subgroups which are not self-normalizing. Rend. Sem. Mat. Univ. Padova 142 (2019), pp. 69–80

DOI 10.4171/RSMUP/30