Is $\mathbb{R/Z}$ isomorphic to $\mathbb{Q/Z}$?

Question

I’ve seen a proof that those quotient groups are isomorphic to the circle group, but I don’t know if they’re isomorphic to each other. By transitivity, they should be, but I cannot prove it directly.

Answer 1

They can't be isomorphic because they have different cardinalities. Indeed, note that $$\mathbb{R} \to S^1: t \mapsto \exp(2 \pi it)$$ induces an isomorphism $\mathbb{R}/\mathbb{Z}\cong S^1$, so $\mathbb{R}/\mathbb{Z}$ is uncountable. On the other hand, $\mathbb{Q}/\mathbb{Z}$ is countable because $\mathbb{Q}$ is countable.

Answer 2

$\Bbb Q/\Bbb Z$ has no infinite cyclic subgroup, whereas $\Bbb R/\Bbb Z$ does.

Answer 3

A group-theoretic answer why they are not isomorphic is that $\mathbb Q/\mathbb Z$ is a torsion group, i.e. each element has a finite order. (The order of $p/q+\mathbb Z$ divides $q$.)

On the other side, some elements in $\mathbb R/\mathbb Z$ are of infinite order (e.g. $\sqrt{2}+\mathbb Z$).