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:
 * $\C_{\ne 0} = \C \setminus \set 0$

The structure $\struct {\C_{\ne 0}, \times}$ is an infinite abelian group.

Proof
Taking the group axioms in turn:

$\text G 0$: Closure
Non-Zero Complex Numbers Closed under Multiplication.

$\text G 1$: Associativity
Complex Multiplication is Associative.

$\text G 2$: Identity
From Complex Multiplication Identity is One, the identity element of $\struct {\C_{\ne 0}, \times}$ is the complex number $1 + 0 i$.

$\text G 3$: Inverses
From Inverse for Complex Multiplication‎, the inverse of $x + i y \in \struct {\C_{\ne 0}, \times}$ is:
 * $\dfrac 1 z = \dfrac {x - i y} {x^2 + y^2} = \dfrac {\overline z} {z \overline z}$

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

$\text C$: Commutativity
Complex Multiplication is Commutative.

Infinite
Complex Numbers are Uncountable.