Subgroup of the unit circle under complex multiplication 
Let $T=\{z:|z|=1 \mathbin{\text{and}} z \in \mathbb C\}$ be the unit circle in the complex plane, considered as a topological group under complex multiplication and the usual topology.
Show that every subgroup of $T$ is either dense in $T$ or finite.

How to prove the denseness?

Hint: The map $\theta \mapsto e^{i \theta}$ is a continuous epimorphism from $(\mathbb{R},+)$ to $(T ,\times)$ and a subgroup of $(\mathbb{R},+)$ is either dense in $\mathbb{R}$ or of the form $\alpha \mathbb{Z}$ for some $\alpha \in \mathbb{R}$.
The above classification of the subgroups of $(\mathbb{R},+)$ is a classical result and can be shown in the following way:


*

*Let $H$ be a subgroup of $(\mathbb{R},+)$ and let $\alpha= \inf \{ h \in H \mid h>0\}$.

*If $\alpha=0$, show that $H$ is dense in $\mathbb{R}$.

*If $\alpha >0$, show that $H= \alpha \mathbb{Z}$.

