Characterization of SL(2,q) by its non-commuting graph

TL;DR

This paper proves that the non-commuting graph uniquely characterizes the group SL(2,q), meaning any group with an isomorphic non-commuting graph to SL(2,q) must be isomorphic to it.

Contribution

It establishes that the non-commuting graph completely determines the structure of SL(2,q) among finite groups.

Findings

Non-commuting graph of SL(2,q) is unique to the group.

Any group with an isomorphic non-commuting graph to SL(2,q) is isomorphic to SL(2,q).

The result characterizes SL(2,q) by its non-commuting graph.

Abstract

Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting graph $\mathcal{A}_G$ of $G$ is the graph whose vertex set is $G\backslash Z(G)$ and two vertices are joined by an edge if they do not commute. Let $\mathrm{SL}(2,q)$ be the special linear group of degree 2 over the finite field of order $q$. In this paper we prove that if $G$ is a group such that $\mathcal{A}_G\cong \mathcal{A}_{\mathrm{SL}(2,q)}$ for some prime power $q\geq 2$, then $G\cong \mathrm{SL}(2,q)$.