Answer to Question #149664 in Abstract Algebra for Sourav Mondal

Question

Let (C*, • ) denote the group of non-zero
complex numbers and
S= {z belongs to C*| IZI = 1 }. Show that
C* / S isomorphic to R+ where (R+, • ) is the group of
positive real numbers.

Answer 1

Let's consider

$\phi : \mathbb{C}^* \to \mathbb{R}^+$

$z\mapsto|z|$

This function is a surjective homomorphism.

For every $x\in\mathbb{R}^+$ , we can view this number as an element of $\mathbb{C}^*$ , $x\in\mathbb{C}^*, |x|=x$ and thus it is surjective.

It is a homomorphism by the properties of complex numbers: $\phi(z_1z_2) = |z_1z_2|=|z_1||z_2|=\phi(z_1)\phi(z_2)$

We can also see that $Ker(\phi) = \{z\in\mathbb{C}^* : \phi(z)=|z|=1 \}=S$ . Thus by the isomorphism theorem $\bar{\phi} : \mathbb{C}^*/S \to \mathbb{R}^+$ is an isomorphism and so $(\mathbb{C}^*/S,\cdot) \overset{\phi}{\simeq} (\mathbb{R}^+,\cdot)$ .