Let $G$ be a transitive subgroup of $S_p$ and let $H$ be a non-trivial normal subgroup of $G$. I need to show that any Sylow p-subgroup of $G$ is also contained in $H$.
I know that any transitive subgroup of $S_p$ contains a non-trivial Sylow $p$ subgroup, of cardinality $p$ and use the Sylow theorems.
Thanks
Since $H$ is normal and all Sylow-$p$ subgroups are conjugate, once you show that $H$ contains one of them, you know that $H$ contains all of them. But since $H$ is transitive, the orbit of every element of $[p]$ is of size $p$ so $p$ divides the order of $H$, and so $H$ contains an element (and thus a subgroup) of order $p$. Thus, $H$ contains all Sylow-$p$ subgroups of $S_p$.