If $H \leqslant G$, $N \trianglelefteq G$, why does $HN=NH$ imply $HN \subseteq \langle H,N \rangle$?

Question

This is related to Geoff Robinson's answer to this question.

Specifically, I don't understand why if $H \leqslant G$, $N \trianglelefteq G$, then $HN=NH$ implies that $HN \subseteq \langle H, N \rangle$, where $\langle H, N \rangle$ is the subgroup of $G$ generated by the union of $H$ and $N$.

Could somebody please explain that inclusion to me? (the other inclusion, $\langle H, N \rangle \subseteq HN$ is obvious, since $\langle H, N \rangle$ is the smallest subset containing both $H$ and $N$.)

Thank you.

Answer 1

It is always true that $HN \subseteq \langle H, N \rangle$ because $\langle H, N \rangle$ contains all products $hn$ (and more).

The hard part is $\langle H, N \rangle \subseteq HN$. For that, you only need to prove that $HN$ is a subgroup.