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

## Theorem

Let $\Q_{\ne 0}$ be the set of non-zero rational numbers:

$\Q_{\ne 0} = \Q \setminus \set 0$

The structure $\struct {\Q_{\ne 0}, \times}$ is a countably infinite abelian group.

## Proof

From the definition of rational numbers, the structure $\struct {\Q, + \times}$ is constructed as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

Hence from Multiplicative Group of Field is Abelian Group, $\struct {\Q_{\ne 0}, \times}$ is an abelian group.

From Rational Numbers are Countably Infinite, we have that $\struct {\Q_{\ne 0}, \times}$ is a countably infinite group.

$\blacksquare$