Semidirect Product with Trivial Action is Direct Product

Theorem
Let $H$ and $N$ be groups.

Let $\Aut N$ denote the automorphism group of $N$.

Let $\phi: H \to \Aut N$ be defined as:
 * $\forall h \in H: \map \phi h = I_N$ for all $h \in H$

where $I_N$ denotes the identity mapping on $N$.

Let $N \rtimes_\phi H$ be the corresponding semidirect product.

Then $N \rtimes_\phi H$ is the direct product of $N$ and $H$.

Proof
Pick arbitrary $\tuple {n_1, h_1}, \tuple {n_2, h_2} \in N \rtimes_\phi H$.

which meets the definition of direct product.