NCF-distinguishablity by prime graph of \(\text{PGL}(2,p)\) where \(p\) is a prime
type: Item
instance of: scholarly article
title: NCF-distinguishablity by prime graph of \(\text{PGL}(2,p)\) where \(p\) is a prime
zbMATH Open document ID: 1246.20032
DOI: 10.1216/RMJ-2011-41-5-1523
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-10-18T00:00:00Z
review text: From the authors' summary: Let \(G\) be a finite group. The prime graph \(\Gamma(G)\) of \(G\) is defined as follows. The vertices of \(\Gamma(G)\) are the primes dividing the order of \(G\) and two distinct vertices \(p\), \(p'\) are joined by an edge if there is an element in \(G\) of order \(pp'\). Let \(p\) be a prime number. We know that \(\text{PGL}(2,p)\) has a unique nonabelian composition factor which is isomorphic to \(\text{PSL}(2,p)\). Assume that \(p\) is not a Mersenne or Fermat prime and \(p\neq 11\), \(19\). In this paper, we determine the structure of finite groups with the same prime graph as \(\text{PGL}(2,p)\) and prove that if \(G\) is a finite group such that \(\Gamma(G)=\Gamma(\text{PGL}(2,p))\) and \(p\neq 13\), then \(G\) has a unique nonabelian composition factor which is isomorphic to \(\text{PSL}(2,p)\) and if \(p=13\), then \(G\) has a unique nonabelian compostion factor which is isomorphic to \(\text{PSL}(2,13)\) or \(\text{PSL}(2,27)\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5959796
zbMATH Keywords: prime graphs
zbMATH Keywords: finite simple groups
zbMATH Keywords: composition factors
zbMATH Keywords: linear groups
zbMATH Keywords: 2-Frobenius groups
author: M. Khatami
author: Behrooz Khosravi
author: Z. Akhlaghi
reviewed by: Rulin Shen
describes a project that uses: GAP
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5959796