A characterization of the finite simple group \(L_{16}(2)\) by its prime graph.
type: Item
instance of: scholarly article
title: A characterization of the finite simple group \(L_{16}(2)\) by its prime graph.
zbMATH Open document ID: 1143.20009
author: Behrooz Khosravi
author: Bahman Khosravi
author: Behnam Khosravi
published in: Manuscripta Mathematica
publication date: 2008-05-27T00:00:00Z
review text: Let \(G\) be a finite group. The prime graph of \(G\), denoted by \(\Gamma(G)\), is a simple graph whose vertex set is the set of all prime divisors \(\Pi(G)\) of \(|G|\), and two distinct vertices \(p,q\) belonging to \(\Pi(G)\) are joined by an edge if \(G\) has an element of order \(pq\). The group \(G\) is said to be recognizable by its prime graph if for every group \(H\) with \(\Gamma(G)=\Gamma(H)\) we have \(G\cong H\). In the paper under review the authors prove that the projective special linear group \(\text{PSL}_{16}(2)\) is recognizable by its prime graph. As a consequence of the main result of this paper the authors deduce that the group \(L_{16}(2)\) is recognizable by the set of its element orders. But it is noteworth to mention that \textit{M. A. Grechkoseeva, W. Shi} and \textit{A. V. Vasilev} [Algebra Colloq. 14, No. 4, 585-591 (2007; Zbl 1139.20012)] proved that the groups \(\text{PSL}_{16}(2^m)\) are recognizable by their sets of element orders. Also \textit{A. V. Zavarnitsine} and \textit{V. D. Mazurov} [Dokl. Math. 74, No. 1, 569-572 (2006); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 409, No. 6, 736-739 (2006; see the following review Zbl 1143.20010)] proved that the groups \(\text{PSL}_n(2)\), \(n\geq 3\), are recognizable by their sets of element orders. Therefore the conjecture mentioned in the paper is answered affirmatively. I would like to mention that this conjecture first was put forward by \textit{M. R. Darafsheh} and \textit{A. R. Moghaddamfar} [Commun. Algebra 29, No. 1, 465-475 (2001; Zbl 0989.20018)].
reviewed by: M. R. Darafsheh
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 5279049
zbMATH Keywords: finite simple groups
zbMATH Keywords: prime graphs
zbMATH Keywords: recognizable groups
zbMATH Keywords: spectra
zbMATH Keywords: projective special linear groups
zbMATH Keywords: sets of element orders
MaRDI profile type: Publication
full work available at URL: s00229-007-0160-9
OpenAlex ID: W1981796029
cites work: Involutions in Chevalley groups over fields of even order
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cites work: CHARACTERIZATION OF THE GROUPSPSL5(2),PSL6(2), ANDPSL7(2)
cites work: Q4415359
cites work: Q5577154
cites work: The Prime Graph of a Sporadic Simple Group
cites work: Prime graph components of the simple groups of Lie type over the field of even characteristic
cites work: Q3544754
cites work: Q5438373
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cites work: Q5755173
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cites work: Q4369487
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cites work: A characterization of the finite simple group \(L_{11}(2)\) by its element orders.
cites work: Q3322282
cites work: A characteristic property ofA 8
cites work: Q4464225
cites work: Prime graph components of finite groups
cites work: Recognition of finite groups by the prime graph
DOI: 10.1007/S00229-007-0160-9
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description: scientific article; zbMATH DE number 5279049