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