Morphism from Multiplicative Group of Complex Numbers to Unit Circle

Theorem

Let $\struct {\C_{\ne 0}, \times}$ denote the multiplicative group of complex numbers.

Let $f: \C_{\ne 0} \to \C_{\ne 0}$ be the mapping defined as:

$\forall z \in \C_{\ne 0}: \map f z = \dfrac z {\cmod z}$

where $\cmod z$ denotes the modulus of $z$.

Then $f$ is an endomorphism on $\struct {\C_{\ne 0}, \times}$ whose kernel is the positive real axis:

$\set {z \in \C: z = x + 0 i, x \in \R_{>0} }$

and whose image is the unit circle:

$\set {z \in \C: \cmod z = 1}$

Proof

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Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{M}$