$(\mathbb{Z}\times\mathbb{Z} \times\mathbb{Z} )/ \langle(3,3,3)\rangle$ is isomorphic to what?

Question

The solution told me it is isomorphic to $\mathbb{Z}_3\times \mathbb{Z} \times \mathbb{Z}$, but why couldn't we argue that $(1,0,0), (0,1,0)$ and $(0,0,1)$ all generates infinite groups, so it is isomorphic to $\mathbb{Z}\times\mathbb{Z} \times\mathbb{Z}$?

Answer 1

This is the way I thought initially.

In vector spaces, if $\{v_1,v_2,v_3\}$ is a basis, then $\{v_1+v_2+v_3,v_2,v_3\}$ is also a basis (since every element in one set can be obtained from elements of other set and these sets have same cardinality).

Similarly, $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ can be considered as group with (basis) generating set $\{(1,1,1), (0,1,0), (0,0,1)\}$ instead of $\{(1,0,0), (0,1,0), (0,0,1)\}$; then the answer to your question is almost clear. I hope this clarifies your question a little and you can fill the details.