Definition:Multiplicative Group of Positive Rational Numbers

Definition

The multiplicative group of positive rational numbers $\left({\Q_{> 0}, \times}\right)$ is the set of (strictly) positive rational numbers under the operation of multiplication.

Also see

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

Thus rational multiplication is:

Well-defined on $\Q_{> 0}$

Closed on $\Q_{> 0}$

Associative on $\Q_{> 0}$

Commutative on $\Q_{> 0}$

The identity of $\left({\Q_{> 0}, \times}\right)$ is $1$

Each element of $\left({\Q_{> 0}, \times}\right)$ has an inverse.