Let $G$ be a group of order $n$, $H$ a subgroup of $G$ of order $m$, $k=\dfrac{n}{m}$ and $S_k$ the symmetric group on $k$ symbols.
(a) Show that there is a non-trivial group homomorphism $\phi : G\to S_k$.
(b) Assuming $\dfrac{k!}{2}<n$, show that $G$ has a non-trivial proper normal subgroup.
I have solved (a) by using the action of $G$ by left-multiplication on the cosets of $H$. But I cannot proceed with (b).
Any help would be appreciated. Thank you!