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
Let $$\frac a b, \frac c d$$ be rational numbers.

From Product of Division Products we observe that $$\frac a b \times \frac c d = \frac {a \times b} {c \times d}$$.

Taking the group axioms in turn:

G0: Closure
Rational Multiplication is Closed.

G1: Associativity
Rational Multiplication is Associative.

G2: Identity
The identity element of $$\left({\Q^*, \times}\right)$$ is the rational number $$\frac 1 1 = 1$$:

G3: Inverses
The inverse of $$\frac p q \in \left({\Q^*, \times}\right)$$ is $$\frac q p$$:

C: Commutativity
Rational Multiplication is Commutative.

Infinite
Rational Numbers are Countable.