Counting the number of centralizers of 2−element subsets in a finite group

Abstract

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 |2−Cent(G)|=n and primitive (2,n)‐centralizer if ∣∣2−Cent(G)∣∣=∣∣2−Cent(GZ(G))∣∣=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≤9, are given.