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.
$\blacksquare$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 3$: Examples of Infinite Groups: $\text{(i)}$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.3$: Example $2$
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.05$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 29 \alpha \ (3)$
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.5$