Isomorphism between complex numbers minus zero and unit circle How do we show that $\mathbb{C}^{\times}$ and $S^{1}$ are isomorphic as groups?

First, note that the additive groups of $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic, since $\mathbb{R}$ and $\mathbb{R}^2$ have the same dimension as vector spaces over $\mathbb{Q}$.
In particular, there exists a group isomorphism $\varphi\colon \mathbb{R} \to \mathbb{R}^2$ such that $\varphi(1) = (1,0)$.  Then $\varphi(\mathbb{Z}) = \mathbb{Z}\times\{0\}$, so
$$
S^1 \;\cong\; \mathbb{R}/\mathbb{Z} \;\cong\; \mathbb{R}^2/(\mathbb{Z}\times\{0\}) \;\cong\; S^1\times\mathbb{R} \;\cong\; \mathbb{C}^\times.
$$
