What is the order and structure of Aut(SL$_2(p)$)?

Question

Here $p$ is a prime. We know that for $Z(SL_2(p)) = \{ \pm I \}$ and $\lvert SL_2(p)\rvert= p^3-p$ so there are $\frac{p^3-p}{2}$ inner automorphisms. What is the outer automorphism group?

Answer 1

It is $\mathrm{PGL}_2(p)$ for $p\geq 5$, which I assume you are thinking about. Any automorphism of $\mathrm{SL}_2(p)$ induces an automorphism on $\mathrm{PSL}_2(p)$. The automorphism group of $\mathrm{PSL}_2(p)$ is $\mathrm{PGL}_2(p)$. The only question is whether one can pull these back to automorphisms of $\mathrm{SL}_2(p)$. You can, and the group $\mathrm{SL}_2(p).2$ is in fact a subgroup of $\mathrm{SL}_2(p^2)$.