prove that the unit circle is isomorphic to the quotient group $\Bbb R/K$, where $K$ is a normal subgroup of $(\Bbb R, +)$

Question

Prove that the circle group $(S,$x$)$

where $S=\{z∈ \Bbb C | |z|= 1\}$

is isomorphic to the quotient group $\Bbb R/K$ where $K$ is a normal subgroup of $(\Bbb R , +)$

I understand the basic definition of isomorphism and how to show it, but I have no idea how to begin to tackle this question. Any hints and help will be appreciated.

Thanks

Answer 1

The subgroup is $K=2\pi \mathbb Z$, and the quotient map sends $\theta\mapsto e^{i\theta}$. Consider this a hint.