Semidirect Product is Abelian iff Components are Abelian and Action is Trivial

Theorem

Let $N$ and $H$ be groups.

Let $H$ act by automorphisms on $N$ via $\phi$.

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

Then the following are equivalent:

$(1): \quad$ $N\rtimes_\phi H$ is abelian
$(2): \quad$ $N$ and $H$ are abelian and $H$ acts trivially

Proof

1 implies 2

Let $n\in N$, $h\in H$.

From $(n,e)(e,h) = (e,h)(n,e)$ we have $n\phi_e(e) = e\phi_h(n)$.

Thus $H$ acts trivially.

By Semidirect Product with Trivial Action is Direct Product, $N\rtimes_\phi H = N\times H$.

By External Direct Product of Abelian Groups is Abelian Group, $N$ and $H$ are abelian.

$\Box$

2 implies 1

By Semidirect Product with Trivial Action is Direct Product, $N\rtimes_\phi H = N\times H$.

By External Direct Product of Abelian Groups is Abelian Group, $N\rtimes_\phi H$ is abelian.

$\Box$

$\blacksquare$