The Quotients of \(G^{3,11,924}\)

Abstract

The groups \(G^{k,l,m}\) have been extensively studied by H. S. M. Coxeter. They are symmetric groups of the maps \(\{k,l\}_m\) which are constructed from the tessellations \(\{k,l\}\) of the hyperbolic plane by identifying two points, at a distance \(m\) apart, along a Petrie path. It is known that \(\text{PSL}(2,q)\) is a quotient group of the Coxeter groups \(G^{(m)}\) if \(-1\) is a quadratic residue in the Galois field \({F}_q\), where \(q\) is a prime power. G. Higman has posed the question that for which values of \(k,l,m\), all but finitely many alternating groups \(A_k\) and symmetric groups \(S_k\) are quotients of \(G^{k,l,m}\). In this paper, we have answered this question by showing that for \(k=3,l=11\), all but finitely many \(A_n\) and \(S_n\) are quotients of \(G^{3,11,m}\), where \(m\) has turned out to be \(924\).