Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers

Theorem

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

The structure $\struct {\Q_{> 0}, \times}$ is a subgroup of $\struct {\Q_{\ne 0}, \times}$, where $\Q_{\ne 0}$ is the set of rational numbers without zero: $\Q_{\ne 0} = \Q \setminus \set 0$.

Proof

From Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group we have that $\struct {\Q_{\ne 0}, \times}$ is a group.

We know that $\Q_{> 0} \ne \O$, as (for example) $1 \in \Q_{> 0}$.

Let $a, b \in \Q_{> 0}$.

Then:

$a b \in \Q_{\ne 0}$ and $ab > 0$

Hence:

$a b \in \Q_{> 0}$

Let $a \in \Q_{> 0}$.

Then:

$a^{-1} = \dfrac 1 a \in \Q_{> 0}$

So, by the Two-Step Subgroup Test, $\struct {\Q_{> 0}, \times}$ is a subgroup of $\struct {\Q_{\ne 0}, \times}$.

$\blacksquare$