Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not. Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.
pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times \mathbb{Z}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \right \}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$. Since $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \right \}$ are isomorphic, $\mathbb{Z}\times \mathbb{Z}/\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}/\mathbb{Z}\times \mathbb{Z}\times \left \{ 0 \right \}$ are isomorphic. But the first one is isomorphic to the trivial group and the second one is isomorphic to $\mathbb{Z}$. It is a contradiction.
Is my proof right? If not, is there another proof?

Suppose  $\mathbb{Z}\times \mathbb{Z}$ is isomorphic to $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$, via a map $\phi$. Then, as $(1,0)$, $(0,1)$ generate  $\mathbb{Z}\times \mathbb{Z}$, $\phi(1,0)$ and $\phi(0,1)$ generate $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$. But  $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ cannot be generated by fewer than 3 elements, a contradiction. So  $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are not isomorphic.
