Constructive Recognition of Special Linear Groups

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.