Abdollahi, Alireza Characterization of \(\text{SL}(2,q)\) by its non-commuting graph. (English) Zbl 1191.20014 Beitr. Algebra Geom. 50, No. 2, 443-448 (2009). Let \(G\) be a non-Abelian group and associate the non-commuting graph \(\mathcal A_G\) with \(G\) as follows: the vertex set of \(\mathcal A_G\) is \(G\setminus Z(G)\) where \(Z(G)\) denotes the center of \(G\) and two vertices \(x\) and \(y\) are adjacent if and only if \(xy\neq yx\). The non-commuting graph \(\mathcal A_G\) was first considered by Paul Erdős in 1975 [see B. H. Neumann, J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]. Let \(\text{SL}(2,q)\) be the special linear group of degree 2 over the finite field of order \(q\). In this paper the author proves that if \(G\) is a group such that \(\mathcal A_G\cong\mathcal A_{\text{SL}(G)}\) for some prime power \(q\geq 2\), then \(G\cong\text{SL}(2,q)\). Reviewer: Shi Wujie (Suzhou) Cited in 1 Document MSC: 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D06 Simple groups: alternating groups and groups of Lie type 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) Keywords:non-commuting graphs; special linear groups; graph isomorphisms Citations:Zbl 0333.05110 PDFBibTeX XMLCite \textit{A. Abdollahi}, Beitr. Algebra Geom. 50, No. 2, 443--448 (2009; Zbl 1191.20014) Full Text: arXiv EuDML EMIS