Definition:Multiplicative Group of Complex Numbers

Definition

The multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$ is the set of complex numbers without zero under the operation of multiplication.

Also see

• Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group

Thus complex multiplication is:

• Well-defined on $\C_{\ne 0}$
• Closed on $\C_{\ne 0}$
• Associative on $\C_{\ne 0}$
• Commutative on $\C_{\ne 0}$
• The identity of $\struct {\C_{\ne 0}, \times}$ is $1$
• Each element of $\struct {\C_{\ne 0}, \times}$ has an inverse.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups