Classify $\mathbb{Z}\times\mathbb{Z}/\langle(0,3)\rangle$ according to the fundamental theorem of finitely generated abelian groups.

Question

I am having an issue classifying $\mathbb{Z}\times\mathbb{Z}/\langle(0,3)\rangle$ according to the fundamental theorem of finitely generated abelian group (i.e. finding what $\mathbb{Z}\times\mathbb{Z}/\langle(0,3)\rangle$ is isomorphic to). I think It should $\mathbb{Z}$, but I am not sure why. Thanks!

Answer 1

Well, it is not $\mathbb{Z}$. What is the order of (the coset) $[(0,1)]$?

You don't need the fundamental theorem at all. What you need is the following: if $G,G'$ are groups and $N\subseteq G$, $N'\subseteq G'$ are normal subgroups then $N\times N'$ is normal in $G\times G'$ and

$$(G\times G')/(N\times N')\simeq (G/N)\times (G'/N')$$

With that you can easily check that $\langle(0,3)\rangle=\{0\}\times 3\mathbb{Z}$ and so your group is $\mathbb{Z}\times\mathbb{Z}_3$.

Answer 2

We have

$\qquad \mathbb{Z}\times\mathbb{Z} = \mathbb{Z} e_1 \oplus \mathbb{Z} e_2 $

$\qquad \langle(0,3)\rangle = \mathbb{Z} (0 e_1) \oplus \mathbb{Z} (3e_2) $

Therefore,

$\qquad \mathbb{Z}\times\mathbb{Z}/\langle(0,3)\rangle \cong \mathbb{Z}\times\mathbb{Z_3}$

An explicit isomorphism is induced by $(x,y) \in \mathbb{Z}\times\mathbb{Z} \mapsto (x, y \bmod 3) \in \mathbb{Z}\times\mathbb{Z_3}$.