Counting the Number of Centralizers of 2-Element Subsets in a Finite Group

Suppose $G$ is a finite group. The set of all centralizers of $2-$element subsets of $G$ is denoted by $2-Cent(G)$. A group $G$ is called $(2,n)-$centralizer if $\mathrm{\mid }2-Cent(G)\mathrm{\mid }=n$ and primitive $(2,n)-$centralizer if $\mathrm{\mid }2-Cent(G)\mathrm{\mid }=\mathrm{\mid }2-Cent(\frac{G}{Z(G)})\mathrm{\mid }=n$, where $Z(G)$ denotes the center of $G$. The aim of this paper is to present the main properties of $(2,n)-$centralizer groups among them a characterization of $(2,n)-$centralizer and primitive $(2,n)-$centralizer groups, $n\le 9$, are given.