Homomorphism from Reals to Circle Group/Corollary
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Theorem
Let $\struct {\R, +}$ be the additive group of real numbers.
Let $\struct {C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.
Let $\phi: \struct {\R, +} \to \struct {C_{\ne 0}, \times}$ be the mapping defined as:
- $\forall x \in \R: \map \phi x = \cos x + i \sin x$
Then $\phi$ is a (group) homomorphism.
Proof
By Euler's Identity, $\phi$ can also be expressed as:
- $\forall x \in \R: \map \phi x = e^{i x}$
From Homomorphism from Reals to Circle Group, $\phi$ is a homomorphism from $\struct {\R, +}$ to the circle group $\struct {K, \times}$.
From Circle Group is Infinite Abelian Group, we note that $\struct {K, \times}$ is a subgroup of the multiplicative group of complex numbers.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{F}$