Constructive Recognition of Special Linear Groups

TL;DR

This paper presents a new constructive recognition algorithm for finite special linear groups that efficiently computes generators, enabling better computational handling within algebra systems like GAP.

Contribution

The paper introduces a novel algorithm for recognizing special linear groups and computing generators, improving performance over previous methods.

Findings

Algorithm outperforms existing methods in efficiency

Successfully implemented in GAP software

Enables efficient word problem solutions

Abstract

We introduce a new constructive recognition algorithm for finite special linear groups in their natural representation. Given a group $G$ generated by a set of $d\times d$ matrices over a finite field $\mathbb{F}_q$, known to be isomorphic to the special linear group $\mathrm{SL}(d,q)$, the algorithm computes a special generating set $S$ for $G$. These generators enable efficient computations with the input group, including solving the word problem. Implemented in the computer algebra system GAP, our algorithm outperforms existing state-of-the-art algorithms by a significant margin. A detailed complexity analysis of the algorithm will be presented in an upcoming publication.