I want to show that $(G,.)$ is a finitely generated abelian group and determine its invariant factor, where $G=\mathbb{Z}^2$ and the binary operation is $(x_1,y_1).(x_2,y_2):=(x_1+x_2,y_1+y_2+x_1x_2)$.
It is easy to show that “$.$” is associative and commutative. The identity element is $(0,0)$, and the inverse is $(-x,x^2-y)$. Thus, $G$ is an abelian group.
Furthermore, $G=\langle (1,0),(0,1) \rangle$, so it is finitely generated.
What I am struggling with is the invariant factors part. I know that it would be something like
$\mathbb{Z}/n_1 \times … \times \mathbb{Z}/n_s \times \mathbb{Z}^r$
Where $n_1,…,n_s$ are the invariant factors and $r$ is the free rank. However, I am not sure how to do that. Any help would be appreciated.