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I don't know whether it is an easy or a hard question, but for the proof of a stronger statement I need to know what the subgroups of $\mathbb{Z}\rtimes\mathbb{Z}=\langle x,y\mid x^{-1}yx=y^{-1}\rangle$ are. (Indeed I only need to find the finitely generated subgroups, but I doubt there are non-finitely generated subgroups.)

At first sight I thought this was easy, since the subgroups of $\mathbb{Z}$ are of the form $n\mathbb{Z}$, but then I realized that the subgroups of a direct product are not the direct product of the subgroups. I read about Goursat's lemma, but this only work for direct products. My question is, the subgroups are of the form $n\mathbb{Z}\rtimes m\mathbb{Z}$? Or there are some other groups?

Thanks for your help.

Marcos
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  • What homomorphism defines the semidirect product? – Shaun May 25 '22 at 17:33
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    @Shaun: Since there are only two automorphisms, and one is trivial, unless this is a direct product it has to be inversion. – Arturo Magidin May 25 '22 at 17:35
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    @Shaun I edited it, but I think that in this case there is only one semidirect product. – Marcos May 25 '22 at 17:36
  • I see now. I'm sorry. Thank you, @ArturoMagidin. – Shaun May 25 '22 at 17:37
  • Related: https://math.stackexchange.com/a/5171/742 – Arturo Magidin May 25 '22 at 17:45
  • @ArturoMagidin that is a nice characterization, thanks! However, are there any easy way to prove it in this case? This is a fairly simple group and I want to avoid using such an strong theorem. – Marcos May 25 '22 at 17:49
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    The group $\Bbb Z\rtimes \Bbb Z$ is isomorphic to the fundamental group of the Klein bottle. So the question is the same as this one. The answer is: "The group is either trivial, free of rank one, free Abelian of rank two, or non-Abelian of rank two." See also this post. – Dietrich Burde May 25 '22 at 18:15

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