Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 50, No. 2, pp. 443-448 (2009)

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Characterization of $\mathrm{SL}(2,q)$ by its non-commuting graph

Alireza Abdollahi

Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran e-mail:

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

Keywords: non-commuting graph, general linear group, special linear group

Classification (MSC2000): 20D60

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Electronic version published on: 28 Aug 2009. This page was last modified: 28 Jan 2013.

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