Direct product of Z9 and Z9
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Contents
Definition
This group is defined as the direct product of two copies of the cyclic group of order nine.
Arithmetic functions
Group properties
Property | Satisfied | Explanation |
---|---|---|
cyclic group | No | |
homocyclic group | Yes | |
elementary abelian group | No | |
metacyclic group | Yes | |
abelian group | Yes | |
group of prime power order | Yes | |
nilpotent group | Yes |
GAP implementation
Group ID
This finite group has order 81 and has ID 2 among the groups of order 81 in GAP's SmallGroup library. For context, there are 15 groups of order 81. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(81,2)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(81,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [81,2]
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 defined using GAP's CyclicGroup and DirectProduct functions:
DirectProduct(CyclicGroup(9),CyclicGroup(9))