Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group

Theorem
Let $\Q_{> 0}$ be the set of strictly positive rational numbers, i.e. $\Q_{> 0} = \left\{{ x \in \Q: x > 0}\right\}$.

The structure $\left({\Q_{> 0}, \times}\right)$ is a countably infinite abelian group.

Proof
From Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers we have that $\left({\Q_{> 0}, \times}\right)$ is a subgroup of $\left({\Q_{\ne 0}, \times}\right)$, where $\Q_{\ne 0}$ is the set of rational numbers without zero: $\Q_{\ne 0} = \Q \setminus \left\{{0}\right\}$.

From Subgroup of Abelian Group is Abelian it follows that $\left({\Q_{> 0}, \times}\right)$ is an abelian group.

From Positive Rational Numbers are Countably Infinite, it follows that $\left({\Q_{> 0}, \times}\right)$ is a countably infinite group.