Cyclic group:Z9

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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

GAP implementation

Group ID

This finite group has order 9 and has ID 1 among the groups of order 9 in GAP's SmallGroup library. For context, there are groups of order 9. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(9,1)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(9,1);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [9,1]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

The group can be described using GAP's CyclicGroup function:

CyclicGroup(9)