Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group

Theorem
Let $\C_{\ne 0}$ be the set of complex numbers without zero, that is:
 * $\C_{\ne 0} = \C \setminus \left\{{0}\right\}$

The structure $\left({\C_{\ne 0}, \times}\right)$ 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 $\left({\C_{\ne 0}, \times}\right)$ is the complex number $1 + 0 i$.

G3: Inverses
From Inverses for Complex Multiplication‎, the inverse of $x + i y \in \left({\C_{\ne 0}, \times}\right)$ is:
 * $\displaystyle \frac 1 z = \frac {x - i y} {x^2 + y^2} = \frac {\overline z} {z \overline z}$

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

C: Commutativity
Complex Multiplication is Commutative.

Infinite
Complex Numbers are Uncountable.