Integer Powers of 2 under Multiplication form Infinite Abelian Group

Theorem
Let $S$ be the set of integers defined as:
 * $S = \set {2^k: k \in \Z}$

Then $\struct {S, \times}$ is an infinite abelian group.

Proof
It can be seen by inspection that $S \subseteq \Q_{>0}$.

That is, all the elements of $S$ are strictly positive rational numbers.

From Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group:
 * $\struct {\Q_{>0}, \times}$ is an infinite abelian group.

It is noted that $S$ is an infinite set and so trivially $S \ne \O$.

Let $a, b \in S$.

Then:
 * $\exists k_1 \in \Z: a = 2^{k_1}$
 * $\exists k_2 \in \Z: b = 2^{k_2}$

We have that:
 * $\dfrac 1 {2^{k_2} } 2^{k_2} = 1$

and so $\dfrac 1 {2^{k_2} } = 2^{-k_2} = b^{-1}$ is the inverse of $b \in \struct {\Q_{>0}, \times}$.

Then we have:

We have that:
 * $k_1 + k_2 \in \Z$

and so $a \times b^{-1} \in \S$.

Hence by the One-Step Subgroup Test, $\struct {S, \times}$ is a subgroup of $\struct {\Q_{>0}, \times}$.

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

Hence by definition $\struct {S, \times}$ is an infinite group.

Finally, from Subgroup of Abelian Group is Abelian, $\struct {S, \times}$ is an abelian group.