# Non-Zero Complex Numbers under Multiplication form 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:

### G0: Closure

$\Box$

### G1: Associativity

$\Box$

### G2: Identity

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

$\Box$

### G3: 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 $z$.

$\Box$

### C: Commutativity

$\Box$

### Infinite

$\blacksquare$