Are the groups $\mathbb R/ \mathbb Z $ and $ \mathbb R^2 / (\mathbb Z \times \{0\} )$ isomorphic?

Question

Is it true that as groups $\mathbb R/ \mathbb Z \cong \mathbb R^2 / (\mathbb Z \times \{0\} )$ ?

I only know that $\mathbb R \cong \mathbb R^2$ (as groups), but I can see no way to decide whether the said quotients are isomorphic or not. Please help. Thanks in advance.

Answer 1

Hint: If $\phi:\mathbb{R}\to\mathbb{R}^2$ is a group isomorphism, what is $\phi(\mathbb{Z})$? In addition, what is the $\mathbb{R}$-dimension of the $\mathbb{R}$-span of $\phi(\mathbb{Z})$?

Answer 2

Yes, they are isomorphic. This follows from the following:

• The groups $(\mathbb{R},+)$ and $(\mathbb{R}^2,+)$ are isomorphic (which you already know);

• For any two nonzero elements $a,b$ of the group $(\mathbb{R},+)$ are "equivalent" in the sense that a group isomorphism of $(\mathbb{R},+)$ sends one to the other. (What's the isomorphism?)

Now $\mathbb{Z}$ in $\mathbb{R}$ is the subgroup generated by $1$, while $\mathbb{Z} \times \{0\}$ in $\mathbb{R}^2$ is the subgroup generated by $(1,0)$. Can you see where the argument goes from here?