# Classifying the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$

We wish to classify the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$, that is, find a group to which it is isomorphic. (According to the fundamental theorem of finitely generated abelian groups. Initially, I thought the group had but two cosets, forcing an isomorphism to $\mathbb{Z}_2$. Obviously, this is wrong, due to the existence of cosets such as $(1, 0) + \langle (2, 2) \rangle$ However, I am unable to see how I am to find an isomorphism here.

• Andrew: my answer was total crap, sorry. – Ian Coley Mar 21 '14 at 8:28
• No problem; I've provided this site with both crappy questions and answers :) – Andrew Thompson Mar 21 '14 at 16:27

• Show that $\Bbb{Z}\times\Bbb{Z}$ is generated (freely) by the elements $u=(1,0)$ and $v=(1,1)$. IOW every element $(a,b)\in\Bbb{Z}\times\Bbb{Z}$ can be written as a linear combination of $u$ and $v$ with integer coefficients in a unique way.
• Show that $mu+nv$ is in the subgroup generated by $(2,2)$ if and only if $m=0$ and $n$ is even.
• Show that the mapping $mu+nv+\langle(2,2)\rangle \mapsto (m,\overline{n})$ is an isomorphism from $(\Bbb{Z}\times\Bbb{Z})/\langle(2,2)\rangle$ to $\Bbb{Z}\times(\Bbb{Z}/2\Bbb{Z})$.
The kernel of the surjective map $\mathbb Z\times\mathbb Z\to \mathbb Z\times\mathbb Z/2\mathbb Z$ given by $(m,n)\mapsto (m-n,n)$ is $\langle(2,2)\rangle$.