A Class of Efficient Presentations of Finite Simple Groups

TL;DR

This paper introduces new, efficient presentations for finite simple groups and related groups, using minimal generators and relations, including applications to hyperbolic groups and sporadic groups.

Contribution

It provides novel presentations of finite simple groups, including Chevalley groups and the Janko group, using minimal generators and relations, with applications to hyperbolic geometry.

Findings

New presentations for finite simple groups and their covers.

Applications to hyperbolic groups for n ≥ 7.

Efficient descriptions of sporadic and Chevalley groups.

Abstract

We exhibit a new presentation of the (equilateral) Von Dyck groups $D(2,3,n), \ n\ge 3$, in terms of two generators of order $n$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $n=3,\, 4,\,5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $n\ge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for $n\ge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain a host of (efficient) presentations of finite simple Chevalley groups of type $A_1$ as well as the sporadic Janko group $J_2$.