Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group

Theorem

Let $\R_{>0}$ be the set of strictly positive real numbers:

$\R_{>0} = \set {x \in \R: x > 0}$

The structure $\struct {\R_{>0}, \times}$ is an uncountable abelian group.

Proof

From Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers we have that $\struct {\R_{>0}, \times}$ is a subgroup of $\struct {\R_{\ne 0}, \times}$, where $\R_{\ne 0}$ is the set of real numbers without zero:

$\R_{\ne 0} = \R \setminus \set 0$

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

Its infinite nature follows from the nature of real numbers.

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$\blacksquare$

Sources

• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.06$