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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.

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    $\begingroup$ $g$ is a generator of $\Bbb{F}_p^\times$, then look at the relations between $\pmatrix{g&0\\0&g},\pmatrix{g&0\\0&g^{-1}},\pmatrix{1&1\\0&1},\pmatrix{1&0\\1&1}$ $\endgroup$
    – reuns
    Oct 28, 2020 at 12:28
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    $\begingroup$ There is a lot of literature on this topic. We have the Steinberg presentations of the classical groups. More recently there has been interest from the computational group theory in short presentations of the finites imple groups. $\endgroup$
    – Derek Holt
    Oct 28, 2020 at 12:30
  • $\begingroup$ @reuns I have seen that general linear groups are generated by two elements over the finite fields, is it true? $\endgroup$
    – vidyarthi
    Oct 28, 2020 at 12:30
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    $\begingroup$ Yes the general linear groups can be generated by two elements. But I think the known short presentations tend to be on slightly larger generating sets - perhaps with four or five generators. See here for example. $\endgroup$
    – Derek Holt
    Oct 28, 2020 at 12:31
  • $\begingroup$ The relations describing $\pmatrix{a&b\\0&c}$ are easy, it is adding $\pmatrix{1&0\\1&1}$ which is a bit a headache $\endgroup$
    – reuns
    Oct 28, 2020 at 12:34

2 Answers 2

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The reference for the 2-element generators mentioned by brett stevens is

D. E. Taylor: Pairs of Generators for Matrix Groups. I, The Cayley Bulletin, 3, 1987, 76-85

https://arxiv.org/abs/2201.09155v1

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  • $\begingroup$ These generators are the ones used by Magma and GAP, so they are in some sense "standard". $\endgroup$ Dec 5, 2023 at 2:59
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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.

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