On \(r\)-recognition by prime graph of \(B_p(3)\) where \(p\) is an odd prime.
type: Item
instance of: scholarly article
title: On \(r\)-recognition by prime graph of \(B_p(3)\) where \(p\) is an odd prime.
zbMATH Open document ID: 1252.20011
author: Zahra Momen
author: Behrooz Khosravi
published in: Monatshefte für Mathematik
publication date: 2012-05-22T00:00:00Z
review text: Let \(G\) be a finite group. The set of all primes dividing \(|G|\) is denoted by \(\pi(G)\). The prime graph \(\Gamma(G)\) of \(G\) is defined as follows: the vertices are the elements of \(\pi(G)\), and two distinct vertices \(p,q\) are joined by an edge if and only if there is an element of order \(pq\) in \(G\). In the present paper, the authors prove that if \(\Gamma(G)=\Gamma(B_p(3))\) where \(p\) is an odd prime, then one of the following holds: (a) \(p>3\) and \(G\cong B_p(3)\) or \(C_p(3)\); (b) \(p=3\) and \(G\cong B_3(3)\), \(C_3(3)\) or \(D_4(3)\), or \(G/O_2(G)\cong\Aut(^2B_2(8))\). -- This result generalizes the result of \textit{R. Shen, W. Shi} and \textit{M. R. Zinov'eva} [Sib. Math. J. 51, No. 2, 244-254 (2010); translation from Sib. Mat. Zh. 51, No. 2, 303-315 (2010; Zbl 1210.20020)] about the recognizability by element orders of the groups considered.
reviewed by: Anatoli Kondrat'ev
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6037508
zbMATH Keywords: finite simple groups
zbMATH Keywords: prime graphs
zbMATH Keywords: recognition
zbMATH Keywords: quasirecognition
zbMATH Keywords: sets of element orders
MaRDI profile type: Publication
full work available at URL: s00605-010-0276-y
OpenAlex ID: W2051070737
cites work: Quasirecognition by prime graph of the simple group \(^2F_4(q)\).
cites work: Q3684278
cites work: A Diophantine equation which arises in the theory of finite groups
cites work: Q5577154
cites work: Decomposition of the Augmentation Ideal and of the Relation Modules of a Finite Group
cites work: Finite simple unisingular groups of Lie type
cites work: RECOGNITION OF SOME FINITE SIMPLE GROUPS OF TYPE Dn(q) BY SPECTRUM
cites work: Prime graph components of the simple groups of Lie type over the field of even characteristic
cites work: Q3999066
cites work: Q4839675
cites work: Quasirecognition by prime graph of simple group D_n(3)
cites work: Quasirecognition by prime graph of \(F_4(q)\) where \(q=2^n>2\).
cites work: Prime graph components of finite almost simple groups
cites work: Q4818192
cites work: Q4369487
cites work: A characterization of some alternating and symmetric groups
cites work: Recognition of simple groups \(B_p(3)\) by the set of element orders.
cites work: Pure quantitative characterization of finite simple groups.
cites work: Certain embeddings among finite groups of Lie type
cites work: An Adjacency Criterion for the Prime Graph of a Finite Simple Group
cites work: Prime graph components of finite groups
DOI: 10.1007/S00605-010-0276-Y
description: scientific article; zbMATH DE number 6037508