Consider the group ($[0; 2\pi)$;$\oplus_{2\pi}$) where $\oplus_{2\pi}$ means addition modulo $2\pi$. Define $S_1$ as the following subset of $\mathbb{C}$
$S_1$ = {$z \in \mathbb{C}; z = e^{i\theta}; \theta \in ([0,2\pi);\oplus_{2\pi})$} ; together with the usual multiplication of complex numbers. Show that this is a group.
I started the problem by substituting $z = e^{i\theta}$ = $cos\theta + isin\theta$. To prove it is a group I decided to try and draw a cayley table. However im not sure how to start calculating this. Any help is appreciated
Hint: It is easy to show that $e^{i\theta_1}\cdot e^{i\theta_2}=e^{i(\theta_1+\theta_2)\pmod{2\pi}}$ (Since, $e^{2i\pi}=1$ ). This would show that the group operation of $S_2$ is binary.
If $e^{i\theta}$ is identity, then $e^{i\theta_1}\cdot e^{i\theta}=e^{i\theta_1}=e^{i(\theta_1+\theta)\pmod{2\pi}}$, can you find what $\theta$ is? Similarly, you can find an expression for inverse (using properties of modulus)
$(\mathbb{C}^*,.)$ is associative and hence any subset of it too.