Multiplicative group of non-zero complex numbers

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

• This group is the set of non-zero complex numbers under multiplication.
• This group is the multiplicative group of the field of complex numbers.

This group is often denoted $({\mathbb{C}}^{\times },\times )$, $(\mathbb{C}{,}^{\times }\cdot )$, or simply as ${\mathbb{C}}^{\times }$.

Group properties

Property	Satisfied	Explanation
Abelian group	Yes	Multiplication of complex numbers is commutative
Nilpotent group	Yes	Abelian implies nilpotent
Simple group	No	Any non-trivial proper subgroup is a normal subgroup since abelian. It has such subgroups, for example multiplicative group of non-zero real numbers

See also

• Multiplicative group of non-zero real numbers
• Multiplicative group of non-zero real algebraic numbers
• Multiplicative group of non-zero rational numbers