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.