On the prime graph of \(\text{PSL}(2,p)\) where \(p>3\) is a prime number.
type: Item
instance of: scholarly article
title: On the prime graph of \(\text{PSL}(2,p)\) where \(p>3\) is a prime number.
zbMATH Open document ID: 1149.20015
author: Bahman Khosravi
author: Behnam Khosravi
author: Behrooz Khosravi
published in: Acta Mathematica Hungarica
publication date: 2008-10-22T00:00:00Z
review text: The prime graph of a group \(G\) is a graph whose vertices are the primes which divide \(|G|\) and \(pq\) is an edge if there is an element \(x\in G\), \(o(x)=pq\). Let \(G=\text{PSL}(2,p)\), \(p\) a prime \(p\equiv 3\pmod 4\) and \(S\) an arbitrary finite group. Assume that \(G\) and \(S\) have isomorphic prime graphs, then the authors show that either \(S=\text{PSL}(2,p)\) or \(p=7\) or 11 and \(G/O_2(G)\cong\text{PSL}(2,8)\) in the first case or \(M_{11}\) in the second. The proof as usual in this area goes via the Kegel-Gruenberg result, which then basically reduces the proof to simple groups \(S\) and then using the classification of the finite simple groups and a case by case analysis gives then the result. At the end of the paper a similar result for \(p\equiv 1\pmod 4\) is stated and a sketch of a proof is given.
reviewed by: Gernot Stroth
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5355806
zbMATH Keywords: finite simple groups
zbMATH Keywords: prime graphs
zbMATH Keywords: element orders
zbMATH Keywords: linear groups
MaRDI profile type: Publication
full work available at URL: s10474-007-6021-x
OpenAlex ID: W2065895100
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DOI: 10.1007/S10474-007-6021-X
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description: scientific article; zbMATH DE number 5355806