I met the following exercise: Let $G$ be a transitive subgroup of $S_p$, where $p$ is a prime and the notion of transitive is defined in the usual sense (action on $\{1,2,...,p\}$). Let $H$ be the subgroup of $G$ generated by elements of $G$ of order $p$. Prove: $H$ is a finite simple group.
So far, I only understood that since $G$ is transitive, its order is divided by $p$, so $G$ contains some $p$-cycles. Then what propositions can I use (I didn't see how to use Sylow) to attack this exercise? And by the way, I'm curious about whether there was any classification of simple subgroups of $S_p$ generated by $p$-cycles? Thanks a lot in advance for any hint or help!