Let $G$ be a simple group which has a maximal subgroup of the form $p:q$ (semidirect product of ${\Bbb Z}_{p}$ and ${\Bbb Z}_{q}$) where $p$ and $q$ are some primes and $q\vert(p-1)$. Also suppose that every Sylow $p$-subgroup of $G$ is ${\Bbb Z}_{p}$ and every Sylow $p$-subgroup is contained properly in exactly one maximal subgroup of $G$.

Can we get that $G\cong L_{2}(p)$.