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

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 $\frac{G}{Z(G)}\cong Z_{p^2}⋊Z_{p^2}$ 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.

29 pages, 5 figures