Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group

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

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

Proof
Taking the group axioms in turn:

G0: Closure
Complex Multiplication is Closed.

G1: Associativity
Complex Multiplication is Associative.

G2: Identity
From Complex Multiplication Identity is One, the identity element of $$\left({\C^*, \times}\right)$$ is the complex number $$1 + 0 i$$.

G3: Inverses
From Inverses for Complex Multiplication‎, the inverse of $$x + i y \in \left({\C^*, \times}\right)$$ is:
 * $$\frac 1 z = \frac {x - i y} {x^2 + y^2} = \frac {\overline z} {z \overline z}$$

where $$\overline z$$ is the complex conjugate of $$z$$.

C: Commutativity
Complex Multiplication is Commutative.

Infinite
Complex Numbers are Uncountable.