Non-Zero Complex Numbers under Multiplication form 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 a group.

Proof

Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

Non-Zero Complex Numbers are Closed under Multiplication.

$\Box$

Group Axiom $\text G 1$: Associativity

Complex Multiplication is Associative.

$\Box$

Group Axiom $\text G 2$: Existence of Identity Element

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

$\Box$

Group Axiom $\text G 3$: Existence of Inverse Element

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 $z$.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.07$
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP: Example $5$