Cyclic group:Z9

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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This group, denoted C_9, \Z_9, \Z/9\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:


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:


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: