Exercise 3, § 2

Show, without using Theorem 1, that the mapping $z \to z/\overline{z}$ is a strict morphism of the topological group $\mathbf{C}^*$ onto the topological group $\mathbf{U}$. Deduce that the mapping $t \to (1 + it)/(1 - it)$ [$(1 + it)/(1 - it) = -1$ if $t = \infty$] is an isomorphism of the topological group $\tilde{\mathbf{R}}$ (no. 6) onto the topological group $\mathbf{U}$, and hence give another proof of Theorem 1.