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
From Non-Zero Complex Numbers under Multiplication form Group, $\struct {\C_{\ne 0}, \times}$ is a group.

Then we have:
 * Complex Multiplication is Commutative

and:
 * Complex Numbers are Uncountable.