Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group

Theorem
Let $\Q^*$ be the set of rational numbers without zero, i.e. $\Q^* = \Q - \left\{{0}\right\}$.

The structure $\left({\Q^*, \times}\right)$ is an infinite abelian group.

Proof
From the definition of rational numbers, the structure $\left({\Q, + \times}\right)$ is constructed as the quotient field of the integral domain $\left({\Z, +, \times}\right)$ of integers.

Hence from Field Product Abelian Group, $\left({\Q^*, \times}\right)$ is an abelian group.

From Rational Numbers are Countable, we have that $\left({\Q^*, \times}\right)$ is countably infinite.

Also see

 * Multiplicative Group of Positive Rational Numbers