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
The identity element of $$\left({\C^*, \times}\right)$$ is the complex number $$1 + 0 i$$:

$$ $$

and similarly:

$$ $$

G3: Inverses
The inverse of $$x + i y \in \left({\C^*, \times}\right)$$ is $$\frac {x - i y} {x^2 + y^2}$$:

$$ $$ $$

Similarly for $$\frac {x - i y} {x^2 + y^2} \left({x + i y}\right)$$.

C: Commutativity
Complex Multiplication is Commutative.

Infinite
Complex Numbers are Uncountable.