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?

David A. Craven · Answer

It is $mathrm{PGL}_2(p)$ for $pgeq 5$, 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 $mathrm{SL}_2(p^2)$.

Answered by David A. Craven on December 15, 2021

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

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