Modulus 1 Rational Argument Complex Numbers under Multiplication form Infinite Abelian Group

Theorem
Let $S$ be the set defined as:
 * $S = \left\{ {\cos \theta + i \sin \theta: \theta \in \Q}\right\}$

Then the algebraic structure $\left({S, \times}\right)$ is an infinite abelian group.

Proof
By definition of polar form of complex numbers, the elements of $S$ are also elements of the circle group $\left({K, \times}\right)$:
 * $K = \left\{ {z \in \C: \left|{z}\right| = 1}\right\}$

$S$ is infinite by construction.

Thus $S \subseteq C$ and trivially $S \ne \varnothing$.

Let $a, b \in S$.

Then:
 * $a = \cos \theta_1 + i \sin \theta_1$

and:
 * $b = \cos \theta_2 + i \sin \theta_2$

for some $\theta_1, \theta_2 \in \Q$.

We have that:

and:

Hence by the Two-Step Subgroup Test, $\left({S, \times}\right)$ is a subgroup of $\left({K, \times}\right)$.

It has been established that $S$ is an infinite set.

Hence by definition $\left({S, \times}\right)$ is an infinite group.

Finally, from Subgroup of Abelian Group is Abelian, $\left({S, \times}\right)$ is an abelian group.