How to show that the circle group T is isomorphic to $\mathbb R/ \mathbb Z$

Question

the circle group is the multiplicative group of all complex numbers of absolute value 1. How can i show that this group is isomorphic with $\mathbb R/ \mathbb Z$. Any hints for the right map is great.

Answer 1

Hint: The complex exponent map.

Answer 2

The map $\phi\colon\Bbb R\to T$ given by $\phi(t)=e^{2\pi it}$ is a group morphism. Hence we can apply first isomorphism theorem.