Abelianization of $\mathbb{Z}\ltimes_\varphi \mathbb{Z}^n$ i would like to ask how to compute the abelianization of the semidirect product $\mathbb{Z}\ltimes_\varphi\mathbb{Z}^n$ where the action is $\varphi(k)v=A^k v$ where $A$ is a fixed invertible matrix in $\mathbb{Z}$. I have read from here a general case < but I don't understand how to particularize to this case. Reading the general case it seems to me that the abelianization would be trivial as $H^{ab}$ is trivial since in my case $H=\mathbb{Z}^n$ is abelian. Thanks!

$H^{ab}=\mathbb{Z}^n$, not trivial. Then you need to take the coinvariants $(H^{ab})_G=\mathbb{Z}^n/\operatorname{image}(A-I)$ Finally take product with $G^{ab}=\mathbb{Z}$, so the end result is $$ \mathbb{Z}\times\operatorname{coim}(A-I). $$