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 $G\cong L\_{2}(p)$.



Characterizing simple groups by their maximal subgroups
The Mathieu group $M\_{23}$ is a counterexample with $p=23$ and $q=11$.


But there are many (almost certainly infinitely many) counterexamples that are semilinear maximal subgroups of classical groups.


As John Shareshian pointed out, for $p=2^k+1$ a Fermat prime, there is an example with $G={\rm PSL}(2,2^k)$ and $q=2$. There are other examples with ${\rm PSL}(2,r)$, such as $r=25$ with $p=13$.


Then for $G={\rm PSL}(3,r)$ we get examples with $q=3$, including $r=3$, $p=13$; $r=5$, $p=31$; $r=7$, $q=19$; etc.


For $G={\rm PSL}(5,2)$ we have an example with $q=31$, $q=5$.


