# Abelian semidirect products

Algebra Level 3

Let $N$ and $H$ be groups, let $\phi : H \to \text{Aut}(N)$ be a homomorphism, and let $G = N \rtimes_\phi H$ be the (outer) semidirect product. Suppose that $G$ is abelian. Which of the following three statements must necessarily be true about $H,N,\phi$?

I. $H$ and $N$ are abelian

II. $\phi$ is the trivial homomorphism

III. $G$ is actually a direct product of groups

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