Recognition of some simple groups by their noncommuting graphs.
type: Item
instance of: scholarly article
title: Recognition of some simple groups by their noncommuting graphs.
zbMATH Open document ID: 1219.20015
author: Liangcai Zhang
author: Shi Wujie
published in: Monatshefte für Mathematik
publication date: 2010-05-17T00:00:00Z
review text: 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}, Characterizat…
reviewed by: A. Iranmanesh
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 5708706
zbMATH Keywords: finite simple groups
zbMATH Keywords: noncommuting graphs
zbMATH Keywords: projective special linear groups
zbMATH Keywords: orders of maximal Abelian subgroups
MaRDI profile type: Publication
full work available at URL: s00605-009-0097-z
OpenAlex ID: W2068778352
cites work: Non-commuting graph of a group.
cites work: Q3684278
cites work: Groups with the same non-commuting graph.
cites work: On finite simple groups of order divisible by three primes only
cites work: Q3440011
cites work: Q3996618
cites work: Q5476748
cites work: A New Characterization ofA10by its Noncommuting Graph
cites work: Prime graph components of finite groups
cites work: Exceptional action of the simple groups \(L_4(q)\) in the defining characteristic.
DOI: 10.1007/S00605-009-0097-Z
Recommended article: Q3572534
Recommended article: Characterization of $ {\mathbb{A}_{16}} $ by a noncommuting graph
Recommended article: Q3625113
Recommended article: Characterization of the alternating group by its non-commuting graph.
Recommended article: Q3110664
Recommended article: A NEW CHARACTERIZATION OF U4(7) BY ITS NONCOMMUTING GRAPH
Recommended article: A characterization of projective special linear group \(L_4(13)\).
Recommended article: Some results on non-commuting graph of a finite group
Recommended article: Non-commuting graph of a group.
Recommended article: Simple groups are characterized by their non-commuting graphs
description: scientific article; zbMATH DE number 5708706