Abelian semidirect products

Algebra Level 5

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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