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

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

2008-08-04

arxiv.org/abs/0808.0377v1

Mathematics

Group Theory

5 pages. to appear in Beitrage zur Algebra und Geometrie

Scientific paper

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 $G\cong \mathrm{SL}(2,q)$.

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