Cyclic group:Z9

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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, denoted $C_9,{\mathbb{Z}}_9,\mathbb{Z}/9\mathbb{Z}$, is defined as the cyclic group of order $9$.

Arithmetic functions

Function	Value	Explanation
order	9
exponent	9
derived length	1
Frattini length	2
Fitting length	1
subgroup rank	1

Group properties

Property	Satisfied	Explanation
cyclic group	Yes
elementary abelian group	No
abelian group	Yes
group of prime power order	Yes
nilpotent group	Yes
solvable group	Yes