Presentations of finite simple groups: a computational approach

TL;DR

This paper provides explicit presentations with bounded relations and bit-length for all nonabelian finite simple groups of rank n over a field of size q, except possibly Ree groups, improving understanding of their algebraic structure.

Contribution

It offers a uniform computational approach to presenting finite simple groups with explicit bounds on relations and bit-length, including classical groups and symmetric groups.

Findings

Finite simple groups have presentations with at most 80 relations.

Symmetric and alternating groups have 3 generators and 7 relations.

Special linear groups have 7 generators and 25 relations.

Abstract

All nonabelian finite simple groups of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, have presentations with at most $80 $ relations and bit-length $O(\log n +\log q)$. Moreover, $A_n$ and $S_n$ have presentations with 3 generators$,$ 7 relations and bit-length $O(\log n)$, while $\SL(n,q)$ has a presentation with 7 generators, $2 5$ relations and bit-length $O(\log n +\log q)$.