The product of a subgroup and a normal subgroup is a subgroup
Solution 1:
Hint. For subgroups $H$ and $K$ of $G$, $HK$ is a subgroup of $G$ if and only if $HK=KH$ as sets.
What happens when one of the two subgroups is normal?
Hint the alternative. For all $a,b,c\in G$, $abc = b(b^{-1}ab)c$.