Groups with the same non-commuting graph.
type: Item
instance of: scholarly article
title: Groups with the same non-commuting graph.
zbMATH Open document ID: 1184.20023
published in: Discrete Applied Mathematics
publication date: 2009-06-30T00:00:00Z
review text: Let \(G\) be a non-Abelian group and associate a non-commuting graph \(\nabla(G)\) with \(G\) as follows: the vertex set of \(\nabla(G)\) 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 \(\nabla(G)\) was first considered by Paul Erdős in 1975 [see \textit{B. H. Neumann}, J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]. It has been conjectured that if \(G\) and \(H\) are two non-Abelian finite groups such that \(\nabla(G)\cong\nabla(H)\), then \(|G|=|H|\) [see \textit{A. Abdollahi, S. Akbari} and \textit{H. R. Maimani}, J. Algebra 298, No. 2, 468-492 (2006; Zbl 1105.20016)]. But this conjecture is not correct [see \textit{A. R. Moghaddamfar}, Sib. Mat. Zh. 47, No. 5, 1112-1116 (2006); translation in Sib. Math. J. 47, No. 5, 911-914 (2006; Zbl 1139.20019)]. Moreover, in the case that \(H\) is a simple group the above conjecture implies \(G\cong H\). In this paper, the author's aim is to prove this case of the conjecture for all the finite non-Abelian simple groups \(H\). Then for certain simple groups \(H\) whose Gruenberg-Kegel graphs are not connected, the author shows that the graph isomorphism \(\nabla(G)\cong\nabla(H)\) implies \(G\cong H\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 5572565
zbMATH Keywords: non-commuting graphs
zbMATH Keywords: finite simple groups
zbMATH Keywords: graph isomorphisms
zbMATH Keywords: Gruenberg-Kegel graphs
author: M. R. Darafsheh
reviewed by: Shi Wujie
MaRDI profile type: Publication
full work available at URL: j.dam.2008.06.010
OpenAlex ID: W1968161690
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DOI: 10.1016/J.DAM.2008.06.010
description: scientific article; zbMATH DE number 5572565