# Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group

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## Theorem

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

The structure $\struct {\Q_{> 0}, \times}$ is a countably infinite abelian group.

## Proof

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

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

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

$\blacksquare$