I've been see the following question on group theory:
Let $p$ be a prime, and let $G = SL2(p)$ be the group of $2 \times 2$ matrices of determinant $1$ with entries in the field $F(p)$ of integers $\mod p$.
(i) Define the action of $G$ on $X = F(p) \cup \{ \infty \}$ by Mobius transformations. [You need not show that it is a group action.]
State the orbit-stabiliser theorem. Determine the orbit of $\infty$ and the stabiliser of $\infty$. Hence compute the order of $SL2(p)$.
I know matrices are isomorphic to Mobius maps, but not how the action of a mobius map can be used to define the action of a matrix (I don't really know what this part means to be honest). I tried the next part, but wasnn't sure whether the consider the vector $(\infty,\infty)$, $(\infty,a)$ or $(b,\infty)$ $(a,b \in F(p))$. Any help would be greatly appreciated!!
(Sorry the question title isn't very related to the question, I just didn't know what to put specifically!)