Normal Subgroup of Subset Product of Subgroups

Theorem
Let $G$ be a group whose identity is $e$.

Let:
 * $H$ be a subgroup of $G$
 * $N$ be a normal subgroup of $G$.

Then:
 * $N \lhd N H$

where:
 * $\lhd$ denotes normal subgroup
 * $N H$ denotes subset product.

Proof
From Subset Product with Normal Subgroup is Subgroup we have that $N H = H N$ is a subgroup of $G$. Note that all elements of $HN$ can be written, by definition, in the form $hn \in HN$.

Take an element $hn \in HN$ and suppose $n_1 \in N$.

Now consider $(hn) n_1 (hn)^{-1} = hn n_1 n^{-1}h^{-1}$. Observe that $n n_1 n \in N$ and $h, h^{-1} \in G$. Then, since N is a normal subgroup of G, we have $(hn) n_1 (n^{-1}h^{-1}) \in N$. Thus $N \lhd NH$.