Isomorphism of groups with complex numbers

Question

I have to prove some isomorphisms of groups, but I am not sure how to write it properly. Let's say we have the groups $G = \{z\in \mathbb{C}^*| |z|^{142} = z^{142}\}$ and $U = \{z\in \mathbb{C}||z|=1\}$

I have to prove that $\mathbb{C}^*/G \cong U$, $G/\mathbb{R}^+ \cong C_{142}$ and $G/C_{142}\cong R^+$

I think I have to define the images $\phi:\mathbb{C}^*\rightarrow U$, $\phi:G\rightarrow C_{142}$ and $\phi:G\rightarrow \mathbb{R}^+$ and use the first isomorphism theorem.

Any ideas how to do that correctly? Thank you in advance!

Answer 1

Hint: Start with the obvious candidates:

$\quad G\rightarrow \mathbb C^*$ given by $z \mapsto \frac{z}{|z|}$

$\quad G\rightarrow R^+$ given by $z \mapsto |z|$

Prove that these maps are group homomorphisms and find their images and kernels.