Non-commuting graph of AC-groups: as matroids

TL;DR

This work examines finite non-abelian groups through their non-commuting graphs $\Gamma_G$, defined on $G\setminus Z(G)$ with edges for noncommuting pairs. It proves a sharp equivalence: $G$ is an $AC$-group (every non-central element has an abelian centralizer) if and only if $\Gamma_G$ is a matroid, linking group structure to matroid properties. A central result provides a direct formula for the clique number $\omega(\Gamma_G)$ via centralizers: if $\{a_1,\dots,a_n\}$ is a maximal set of pairwise non-commuting elements, then $\omega(\Gamma_G)=n$ and $|G|=(1-\omega(\Gamma_G))|Z(G)|+\sum_{i=1}^{\omega(\Gamma_G)}|C_G(a_i)|$. The paper also yields corollaries about subgroups and extends the framework to $CC$-groups and certain $p$-groups, with concrete examples showing $S_n$ and $A_n$ are not AC (hence their $\Gamma_G$ are not matroids) while dihedral groups $D_{2n}$ yield matroid graphs, illustrating the theory's reach.

Abstract

Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph ΓG (called non-commuting graph of G) as follows: Take G\Z(G) as the vertices of ΓG 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

Theorems & Definitions (9)

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