New efficient presentations for \(\text{PSL}(2,5)\) and \(\text{SL}(2,5)\); the structure of the groups \(G(5,m)\) and \(G(m,n,r)\)
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 \(r=(3m-2n+7)\).
scientific article; zbMATH DE number 897258