On the clique number of non-commuting graphs of certain groups

TL;DR

This paper characterizes non-solvable groups with a non-commuting graph clique number up to 57, specifically relating to the group $ ext{PSL}(2,7)$, and completes the classification for all finite minimal simple groups.

Contribution

It provides a complete characterization of non-solvable groups with a non-commuting graph clique number at most 57, extending the understanding of the structure of these groups.

Findings

Identified all non-solvable groups with clique number ≤ 57.

Determined the clique number for all finite minimal simple groups.

Connected the clique number to the structure of specific simple groups.

Abstract

Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joint if and only if they do not commute. In a finite simple graph $\Gamma$ the maximum size of a complete subgraph of $\Gamma$ is called the clique number of $\Gamma$ and it is denoted by $\omega(\Gamma)$. In this paper we characterize all non-solvable groups $G$ with $\omega(\mathcal{A}_G)\leq 57$, where the number 57 is the clique number of the non-commuting graph of the projective special linear group $\mathrm{PSL}(2,7)$. We also complete the determination of $\omega(\mathcal{A}_G)$ for all finite minimal simple groups.