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Characterization of \(\text{SL}(2,q)\) by its non-commuting graph. (English) Zbl 1191.20014

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)

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.)

Citations:

Zbl 0333.05110
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