New efficient presentations for \(\text{PSL}(2,5)\) and \(\text{SL}(2,5)\); the structure of the groups \(G(5,m)\) and \(G(m,n,r)\)
type: Item
instance of: scholarly article
title: New efficient presentations for \(\text{PSL}(2,5)\) and \(\text{SL}(2,5)\); the structure of the groups \(G(5,m)\) and \(G(m,n,r)\)
zbMATH Open document ID: 0861.20037
DOI: 10.1007/BF00052109
author: Bilal Vatansever
published in: Acta Mathematica Hungarica
publication date: 1997-05-13T00:00:00Z
review text: Let \(M(F)\) denote the Schur multiplier of a group \(F\). Then \(F\) with \(n\) generators is said to have efficient presentation if it has \(n+\text{rank}(M(F))\) relations. In the paper under review the author uses a nice group-theoretic argument supplemented with CAYLEY program to prove that \(G(5,m)=\langle a,b: a^5=(ab)^2=b^{m+5}a^{-m}b^ma^{-m}=1\rangle\) is isomorphic to \(\text{PSL}(2,5)\) and it has efficient presentation provided \(m\equiv 2\pmod 5\) or \(m\equiv 3\pmod 5\). He shows also that \(G(5,m)\) is trivial if \(m\equiv 1\pmod 5\) or \(m\equiv 4\pmod 5\) and it is infinite if \(m\equiv 0\pmod 5\). In the end he proves also that \(G(m,n,r)=\langle a,b:a^{-5}(ab)^2=1\), \(b^{m+5}a^{-n}b^ma^{-n}=a^r\rangle\) is \(\text{SL}(2,5)\) and it has efficient presentation for \(m\equiv 2\) or \(3\pmod 5\), \(n\equiv 2\) or \(3\pmod 5\) and \(r=(3m-2n+8)\) of \(r=(3m-2n+7)\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 897258
zbMATH Keywords: \(\text{SL}(2,5)\)
zbMATH Keywords: Schur multiplier
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: \(\text{PSL}(2,5)\)
zbMATH Keywords: efficient presentations
reviewed by: Qaiser Mushtaq
describes a project that uses: Cayley
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 897258