Recognition of some simple groups by their noncommuting graphs.
Let \(G\) be a group and \(Z(G)\) be its center. We will associate a graph \(\Gamma(G)\) to \(G\) which is called the `non-commuting graph' of \(G\). The vertex set \(V(\Gamma(G))\) is \(G-Z(G)\) and the edge set \(E(\Gamma(G))\) consists of \((x,y)\), where \(x\) and \(y\) are distinct non-central elements of \(G\) such that \(xy\neq yx\). According to \textit{B. H. Neumann}, [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)], the non-commuting graph of a finite group \(G\) was first considered by P. Erdős in connection with the following problem: Let \(G\) be a group whose non-commuting graph \(\Gamma(G)\) has no infinite complete subgraphs. Is it true that there is a finite bound on the cardinalities of complete subgraphs of \(\Gamma(G)\)? In fact, Neumann gave a positive answer to Erdős's question and this was the origin of many similar research about the non-commuting graph of a group. Recently \textit{A. Abdollahi, S. Akbari} and \textit{H. R. Maimani}, [in J. Algebra 298, No. 2, 468-492 (2006; Zbl 1105.20016)], studied graph properties of the non-commuting graph associated to a non-Abelian group. In particular the authors put forward the following conjecture. Conjecture (AAM Conjecture). Let \(S\) be a finite non-Abelian simple group and let \(G\) be a group such that \(\Gamma(S)\cong\Gamma(G)\). Then \(S\cong G\). In several papers, it has been proved that the AAM Conjecture is true for the finite simple groups with disconnected prime graph, and the alternating group \(A_{10}\) [\textit{M. R. Darafsheh}, Discrete Appl. Math. 157, No. 4, 833-837 (2009; Zbl 1184.20023) and \textit{L. Wang} and \textit{W. Shi}, Commun. Algebra 36, No. 2, 523-528 (2008; Zbl 1153.20013)] which has connected prime graph. In the paper under review, the authors prove that the AAM Conjecture is also true for the projective special linear group \(L_4(9)\) with connected prime graph. Remark: Neda Ahanjideh, in part of her Ph.D. thesis [\textit{N. Ahanjideh}, Characterization of some finite groups by order of their normalizer of Sylow subgroups, University of Tarbiat Modares, Tehran, January 2009], showed that for a given arbitrary non-Abelian group \(H\) and an arbitrary finite non-Abelian simple group \(S\), if \(\Gamma(H)\cong\Gamma(S)\), then \(H\) and \(S\) have the same set of orders of maximal Abelian subgroups, and \(\text{GK}(H)=\text{GK}(S)\) where \(\text{GK}(H)\) is the prime graph of \(H\). In particular, she showed that \(\text{PSL}(3^k+1,2)\) is uniquely determined by the set of orders of its maximal Abelian subgroups, equivalently, by its non-commuting graph, where \(2\mid k\), \(4\nmid k\) and \(k\geq 3\). This new method in her thesis also works for all finite non-Abelian simple groups with connected prime graph or disconnected prime graph.
scientific article; zbMATH DE number 5708706
scientific article