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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.2$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(1)$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(ii)}$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.5$