I am looking for a criterion for a group to be isomorphic to $PSL(2,p^k)$ in terms of its Sylow $p$-subgroups.
For example, let $G$ be finite group of order $p^km$ where $p$ is an odd prime not dividing $m$. Suppose that
- $G$ is simple.
- Sylow $p$-subgroups of $G$ are elementary abelian.
- Any two distinct Sylow $p$-subgroups of $G$ intersect trivially.
- $G$ has exactly $p^k+1$ Sylow $p$-subgroups.
Do these conditions imply that $G\cong PSL(2,p^k)$? (Presumably, such an isomorphism would be proved by treating the $p^k+1$ Sylow $p$-subgroups as the $p^k+1$ points of projective space).
Is there similar set of conditions that implies that $G\cong PSL(2,p^k)$?