Non-Zero Complex Numbers under Multiplication form Abelian Group

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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

Non-Zero Complex Numbers Closed under Multiplication.

$\Box$


G1: Associativity

Complex Multiplication is Associative.

$\Box$


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

Complex Multiplication is Commutative.

$\Box$


Infinite

Complex Numbers are Uncountable.

$\blacksquare$


Sources