Non-commuting graph of AC-groups: as matroids

TL;DR

This paper characterizes AC-groups through their non-commuting graphs, showing these graphs are matroids, and provides a formula for their clique number, deepening understanding of their structure.

Contribution

It establishes that a finite group is an AC-group if and only if its non-commuting graph is a matroid, linking group theory with matroid theory.

Findings

Non-commuting graphs of AC-groups are matroids.

A formula for the clique number of these graphs is derived.

Characterization of AC-groups via graph properties.

Abstract

Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph {\Gamma}G (called non-commuting graph of G) as follows: Take G\Z(G) as the vertices of {\Gamma}G and join x and y, whenever $xy \not= yx$. In this paper, we show that a finite group G is an AC-group, if and only if, the associated non-commuting graph of G is a matroid. Leveraging the properties of matroids, we further delve into the characteristics of AC-groups. Additionally, we provide a formula to compute the clique number of the non-commuting graph of AC-groups, offering a new perspective on the structure of these groups