Subgroup of circle group

Question

If we look at the circle group: $C:=\{ z \in \mathbb{C} : |z| = 1 \}$, I was wondering what a subgroup of $C$ in which each element has a finite order ($\{ z\in C \:| \: \operatorname{ord}(z)<\infty \}$) would look like?

Answer 1

That group is $G=\{e^{i\pi q}\mid q\in\Bbb Q\}$. It is divisible, i.e. if $g\in G$ and $n\in\Bbb N$, then there is a $h\in G$ such that $h^n=g$. Another way of describing it is $G=\{z\in S^1\mid(\exists n\in\Bbb N):z^n=1\}$.

Answer 2

Maybe consider a cyclic subgroup $\langle g \rangle$. Just find a generator $g \in C$ such that $g^n=1$ for some finite $n$. For example, the $n$th roots of unity.