How do we determine the generators and relations for general / special linear groups over a finite field? I will be particularly interested in prime fields, in that also for $\mathbb{Z}_2$?
I know that these groups are a subgroup of $S_{p^n-1}$, the symmetric group. But, apart from that can we derive the generators and relations using direct matrices? Thanks beforehand.
In his review of the paper Waterhouse, William C. "Two generators for the general linear groups over finite fields", Linear and Multilinear Algebra 24 (1989), no. 4, pp.227–230, for AMS MathSciNet, François Digne mentions that in John J. Canon's Cayley algebra system the General Linear Group over finite field $\mathbb{F}$ is generated by $diag(\alpha,1,⋯,1)$ and $(-E_{1,1} -E_{1,2} -E_{2,3}- \cdots -E_{n−1,n} +E_{n,1})$, were $\alpha$ is a primitive element of $\mathbb{F}$ and $E_{r,c}$ is the matrix with a 1 in position $(r,c)$ and zeros everywhere else. Additionally the Special Linear Group is generated by $diag(\alpha,\alpha^{-1},⋯,1)$ and $(-E_{1,1}-E_{1,2}-E_{2,3} - \cdots -E_{n−1,n}+E_{n,1})$.
I realize this is only part (just the generators) of the answer you are looking for, but perhaps it gets you a bit farther.
I am trying to track down a reference for this solution which is better than a review of an article.