Multiplicative group of non-zero complex numbers: Difference between revisions
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|[[Satisfies property::Nilpotent group]] || Yes || [[Abelian implies nilpotent]] | |[[Satisfies property::Nilpotent group]] || Yes || [[Abelian implies nilpotent]] | ||
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|[[Satisfies property::Simple group]] || No || | |[[Satisfies property::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]] | ||
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Revision as of 17:00, 9 January 2024
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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.
Group properties
| Property | Satisfied | Explanation |
|---|---|---|
| Abelian group | Yes | Cyclic implies abelian |
| 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 |