Positive Real Axis forms Subgroup of Complex Numbers under Multiplication

Theorem

Let $S$ be the subset of the set of complex numbers $\C$ defined as:

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

That is, let $S$ be the positive real axis of the complex plane.

Then the algebraic structure $\struct {S, \times}$ is a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.

Proof

We have that $S$ is the same thing as $\R_{>0}$, the set of strictly positive real numbers:

$\R_{>0} = \set {x \in \R: x > 0}$

From Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group, $\struct {S, \times}$ is a group.

Hence as $S$ is a group which is a subset of $\struct {\C_{\ne 0}, \times}$, it follows that $\struct {S, \times}$ is a subgroup of $\struct {\C_{\ne 0}, \times}$.

$\blacksquare$

Sources

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