Let $G$ be a group, $H$ a subgroup of $G$, and $N$ a normal subgroup of $G$.
Verify that $HN=\{hn\mid h \in H, n \in N\}$ is a subgroup of $G$.
Let $G$ be a group, $H$ a subgroup of $G$, and $N$ a normal subgroup of $G$.
Verify that $HN=\{hn\mid h \in H, n \in N\}$ is a subgroup of $G$.
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 $a,b,c\in G$, $abc = b(b^{-1}ab)c$.