On $n$-centralizer $CA$-groups

TL;DR

This paper characterizes finite non-abelian groups with a specific number of centralizers, showing that such groups have orders related to powers of two, especially when they are $CA$-groups or have a derived subgroup of order two.

Contribution

It establishes new conditions on the order of $m$-centralizer groups, specifically identifying when $m$ is a power of two, for both $CA$-groups and groups with a derived subgroup of order two.

Findings

If $G$ is a finite non-abelian $m$-centralizer $CA$-group, then $m=2^r$ for some $r>1.

If $G$ is an $m$-centralizer non-$CA$-group with $|G'|=2$, then $m=2^{2s}$ for some $s>1.

The structure of $G$ is heavily constrained by the size of its center and derived subgroup.

Abstract

Let $G$ be a finite non-abelian group and $m=|G|/|Z(G)|$. In this paper we investigate $m$-centralizer group $G$ with cyclic center and we will prove that if $G$ is a finite non-abelian $m$-centralizer $CA$-group, then there exists an integer $r>1$ such that $m=2^r.$ It is also prove that if $G$ is an $m$-centralizer non-abelian finite group which is not a $CA$-group and its derived subgroup $G'$ is of order 2, then there exists an integer $s>1$ such that $m=2^{2s}.$