In Presentation of SL$(n,\mathbb{Z}_p)$, it is asked whether there are known presentations of $SL(n,p)$. Its comments (particularly this one) and current answer hint at the existence of such presentations.
It is stated that that question is too broad. Not to worry:
The Question:
What are some presentations of $SL(2,q)$?
Further stipulations:
Just in case it is too broad, here are some ways I could narrow the interest:
- I am interested in when $q\ge 4$.
- If a presentation could lend itself well to studying conjugates in $SL(2,q)$, I would prefer it.
- Presentations that are easy to implement in GAP for small values of $n,q$ are preferred.${}^\dagger$
Context:
I know of this (pdf) for $SL(2,p)$. I haven't read it in detail.
Further Context:
- What are you studying?
Essentially, as part of a research degree, I'm looking at properties of a function called $\Delta$ when applied to finite groups of Lie type; $SL(2,q)$ is such a group.
- What text is this drawn from, if any? If not, how did the question arise?
No text.
It seems natural to draw upon my experience with group presentations in past research, for my current research.
- What kind of approaches (to similar problems) are you familiar with?
Nothing in particular. Here is a link to an MSE search of all my combinatorial-group-theory posts. There's 71 so far.
- What kind of answer are you looking for? Basic approach, hint, explanation, something else?
I'm hoping for a short list of presentations of $SL(2,q)$, preferably with references.
- Is this question something you think you should be able to answer? Why or why not?
Given enough time, yes. But this question is simply a bud on the tree of my current research project. I would appreciate some green fingers to help it grow!
$\dagger$: I realise that I could reverse engineer a presentation for small values of $n,q$ in GAP, but I hope to work more generally than that. If I could avoid having to reproduce known presentations from limited data, I would.