Counting Centralizers of a Finite Group with an Application in Constructing the Commuting Conjugacy Class Graph

TL;DR

This paper investigates the structure of centralizers in a specific non-abelian finite group and determines the exact number of element centralizers, also analyzing the structure of its commuting conjugacy class graph.

Contribution

It characterizes the centralizers in a non-abelian finite group with a specific quotient structure and determines the commuting conjugacy class graph.

Findings

The group has exactly [(p+1)^2+1] element centralizers.

The structure of the commuting conjugacy class graph is fully determined.

Provides insights into the relationship between group structure and centralizer properties.

Abstract

The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property that $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a consequence, it is proved that such a group has exactly $[(p+1)^2+1]$ element centralizers and the structure of the commuting conjugacy class graph of $G$ is completely determined.