Definition:Multiplicative Group of Rational Numbers

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

Also see

 * Rational Numbers under Multiplication form 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 $\left({\Q_{\ne 0}, \times}\right)$ is $1$
 * Each element of $\left({\Q_{\ne 0}, \times}\right)$ has an inverse.