Help with semidirect product I need help with this problem, i am trying to understand the semidirect product, so if anyine could help or give me some ideas
Let $G$ be the group generated by $<a,b>$ and the relations $aba^{-1}b=1$.


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*Prove that G has a normal subgroup $H\simeq\mathbb{Z}$ such that $G/H\simeq\mathbb{Z}$

*Show that exist a short exact sequence of the form
$$0 \rightarrow \mathbb{Z}\oplus\mathbb{Z} \rightarrow G \rightarrow \mathbb{Z}_2 \rightarrow 0$$

*Show that $G$ is isomorphic to only one of the next groups:
$$\mathbb{Z} \times \mathbb{Z} ; \mathbb{Z} \times \mathbb{Z}_2;  \mathbb{Z}\ltimes\mathbb{Z} ;  \mathbb{Z}\ltimes\mathbb{Z}_2$$

Notice that $aba^{-1}=b^{-1}$. This means that the cyclic subgroup $<a>$ action (by conjugation) on the cyclic subgroup $<b>$ is an automorphism. Now think about the definition of semidirect product. Which group is acting on which group, and in what way, in a semidirect product? Specifically, if $G=H\ltimes_{\Phi} N$, and if you take $h\in H$ and $n\in N$, what would you expect to be the result of $hnh^{-1}$, written in term of $\Phi$?
