Let $G$ be a finite group, and let $H$ be a proper subgroup of $G$ of index $n$.

Show that there is a normal subgroup $K$ of $G$ such that $|G / K|$ divides $n$ ! and $|G / K| \geqslant n$.

Show that if $G$ is non-abelian and simple, then $G$ is isomorphic to a subgroup of $A_{n}$.