On finite groups with exactly two non-abelian centralizers

TL;DR

This paper characterizes finite groups with exactly two non-abelian centralizers, revealing structural properties and relationships between centralizers, the Fitting subgroup, and the commutator subgroup.

Contribution

It provides a characterization of finite groups with a unique proper non-abelian centralizer, improving previous results and establishing new structural relationships.

Findings

If $C(a)$ is the proper non-abelian centralizer, then $C(a)/Z(G)$ is the Fitting subgroup of $G/Z(G)$.

The centralizer $C(a)$ is the Fitting subgroup of $G$, and $G'$ is contained in $C(a)$.

The group $G$ has exactly two non-abelian centralizers, with specific subgroup relationships.

Abstract

In this paper, we characterize finite group $G$ with unique proper non-abelian element centralizer. This improves \cite[Theorem 1.1]{nab}. Among other results, we have proved that if $C(a)$ is the proper non-abelian element centralizer of $G$ for some $a \in G$, then $\frac{C(a)}{Z(G)}$ is the Fitting subgroup of $\frac{G}{Z(G)}$, $C(a)$ is the Fitting subgroup of $G$ and $G' \in C(a)$, where $G'$ is the commutator subgroup of $G$.