2.10.6

We will first describe the cosets of $$H = \{ \pm 1, \pm i \}$$ in $$G = \mathbb C^\times.$$ Let $$x \in \mathbb C^\times$$, then


 * $$1x$$ does nothing,
 * $$-1x$$ rotates $$x$$ by $$\pi,$$
 * $$ix$$ rotates $$x$$ by $$\frac{\pi}{2},$$
 * $$-ix$$ rotates $$x$$ by $$-\frac{\pi}{2}.$$

Therefore, the coset $$xH$$ is the set of $$x$$ rotated by $$\frac{\pi}{2}$$ (around the origin) any number of times.

To prove that $$G/H$$ is isomorphic to $$G$$ we use the surjective function $$f(a e^{\alpha i}) = a e^{4(\alpha \bmod \frac{\pi}{2}) i}.$$ $$f(a e^{\alpha i} b e^{\beta i}) = ab e^{4(\alpha + \beta \bmod \frac{\pi}{2}) i}$$ and $$f(a e^{\alpha i})f(b e^{\beta i}) = ae^{4(\alpha \bmod \frac{\pi}{2}) i} be^{4(\beta \bmod \frac{\pi}{2}) i} = abe^{4(\alpha + \beta \bmod \frac{\pi}{2}) i}$$, i.e. $$f(xy) = f(x)f(y)$$ where $$x$$, $$y$$ belongs to $$\mathbb C^\times.$$ By the first isomorphism theorem, $$G/H$$ is isomorphic to $$\mathbb C^\times.$$