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Cyclic group:Z9


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

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: