Trouble to solve that $\mathbb C^*$ is isomorphic to $\mathbb R^+ \times {\mathbb R}/{\mathbb Z}$?

Question

How can I show that $\mathbb C^*$ is isomorphic to $\mathbb R^+ \times {\mathbb R}/{\mathbb Z}$?

I am unable to get the required isomorphism.Please help me.

Answer 1

Assuming our groups are $(\mathbb{C}^*, \cdot)$, $(\mathbb{R}^+, \cdot)$ and $(\mathbb{R}/\mathbb{Z}, +)$ (the group operations must be clear), we can have $\phi: \mathbb{C}^* \to \mathbb{R} \times \mathbb{R}/\mathbb{Z} $ be such that: $$\phi(z) = (|z|, \frac{arg(z)}{2\pi})$$ We have $$\phi(zw) = (|zw|, \frac{arg(z) + arg(w)}{2\pi}) = \phi(z)\phi(w)$$ And $$\phi^{-1}(r, \alpha) = re^{2\pi\alpha i}$$ From which follows $\phi$ is an isomorphism.