Definition:Multiplicative Group of Rational Numbers

Definition

The multiplicative group of rational numbers $\struct {\Q_{\ne 0}, \times}$ is the set of rational numbers without zero under the operation of multiplication.

Also see

• Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group

Thus rational multiplication is:

• Well-defined on $\Q_{\ne 0}$
• Closed on $\Q_{\ne 0}$
• Associative on $\Q_{\ne 0}$
• Commutative on $\Q_{\ne 0}$
• The identity of $\struct {\Q_{\ne 0}, \times}$ is $1$
• Each element of $\struct {\Q_{\ne 0}, \times}$ has an inverse.