Abelian semidirect products

Algebra Level 3

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

I. HH and NN are abelian

II. ϕ\phi is the trivial homomorphism

III. GG is actually a direct product of groups


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