I am interested in finding all the relations between two matrices in $SL(2, \mathbb{F}_3)$. Namely, these matrices are $$\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$$ I tried and it looks painful to me. I am not sure at which moment I have to stop. What is the best way to do that? Is it possible to do it in SAGE? My strategy was to just write down all the words in these two matrices and see what happens but I don't this it's a good idea.
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Denote these matrices by $A$ and $B$. Since $A^3=I$, it is enough to compute $A^2$. Since $B^4=I$, compute $B^2$ and $B^3$. Now look at products between $A,A^2$ and $B,B^2,B^3$. The group has $24$ elements. By the way, generators are also given at this duplicate. – Dietrich Burde Sep 10 '23 at 12:47
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@DietrichBurde but I'm interested in these particular generators. Also, they don't explain all the relations between those two generators that you linked so I'm not sure how it helps. It makes sense that it's enough to consider only small powers but I'm not sure how to bound from above the number of relations. – iou Sep 10 '23 at 16:44
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Taking an isomorphism from the presentation given at the other posts, mapping your generators to the others, or vice versa, it is clear that there will not be many relations. You can take the standard presentation here. It has only $6$ relations. – Dietrich Burde Sep 10 '23 at 16:56
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@DietrichBurde This link is useful, thanks! – iou Sep 10 '23 at 17:03
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@DietrichBurde But it seems to me it contains a mistake. Namely, those matrices that they give as generators at bottom do not satisfy cac^(-1)=b. – iou Sep 10 '23 at 19:00
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@DietrichBurde Never mind, I figured it out, thanks! – iou Sep 11 '23 at 21:01