Can $PSL(2, p)$ be a subgroup of $PSL(2,q)$ if $p$ doesn't divide $q$?

Question

Let $p$ be a prime and let $q$ be a power of another prime. Suppose $p\nmid q$, then is it possible that $PSL(2, p)$ can be isomorphic to a subgroup of $PSL(2,q)$?

Any suggestion or reference is welcome.

Answer 1

This can happen if $p\in \{2,3,5\}$.

Look here for a list of the subgroups of $\mathrm{PSL}(2,q)$: https://www.staff.ncl.ac.uk/o.h.king/KingBCC05.pdf

You will see for example that $\mathrm{PSL}(2,5)(\cong A_5)$ occurs when $q\equiv \pm 1 \pmod {10}$.

You can do the same for $p\in\{2,3\}$ by remembering that $\mathrm{PSL}(2,3)\cong A_4$ and $\mathrm{PSL}(2,2)$ is a dihedral group of order $6$.