Problems understanding a statement about the generators of $SL_2(\mathbb{F}_7)$ and $SL_2(\mathbb{F}_p)$

Question

Well, I'm having some problem understanding a statement of a book (I won't provide a screenshot being this book not in English) that state: $SL_2(\mathbb{F}_7)$ is generated $\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$ and $\begin{pmatrix}1 & 0\\ 1 & 1\end{pmatrix}$

In which sense generated? (I suppose multiplicatively, but I'll love to have a confirmation).

It's also true the same thing for $SL_2(\mathbb{F_p})$ for each prime $p$ ?

How to prove all this?

If you have the patient to write an answer I'll be very happy, but even a reference is enough.

Answer 1

$SL_2(\mathbb{Z})$ is generated by

$$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} $$

and the groups you are asking about are images of this group. This classical result has a number of proofs: my preference is to let $SL_2(\mathbb{Z})$ act on the Farey tree and study stabilizers.

Answer 2

I will post a HINT as an answer that might help you to find on your own the right way to prove it. There is a proof of the above explicitly written in Lang's - Algebra (check the 3rd edition, chapter XIII, Lemma 8.1) (at least I read that there for a first time). Think that the above matrices when acting on $SL_2(\mathbb{F})$, no matter field you have, they are acting as elementary row-column operations. Hence the problem can be reduced into finding the solutions of some equations, when you want to split up an element of $SL_2(\mathbb{F})$ in terms of your generating set. For any queries if you don't have the book (or cannot find it) please do let me know.