Multiplicative Group of Complex Numbers

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.

$\Box$

G1: Associativity

Complex Multiplication is Associative.

$\Box$

G2: Identity

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

$\Box$

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 $z$.

$\Box$

C: Commutativity

Complex Multiplication is Commutative.

$\Box$

Infinite

Complex Numbers are Uncountable.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.2$
• Ian D. Macdonald: The Theory of Groups (1968): $\S 1$: Example $1.07$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (1)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.5$