Direct product of Z8 and Z8
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Contents
Definition
This group can be defined as the external direct product of two copies of the cyclic group of order eight. In other words, it has the presentation:
.
As an abelian group of prime power order
This group is the abelian group of prime power order corresponding to the partition:
In other words, it is the group .
Value of prime number ![]() |
Corresponding group |
---|---|
generic prime | direct product of cyclic group of prime-cube order and cyclic group of prime-cube order |
3 | direct product of Z27 and Z27 |
5 | direct product of Z125 and Z125 |
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 64#Arithmetic functions
Group properties
Property | Satisfied? | Explanation |
---|---|---|
cyclic group | No | |
homocyclic group | Yes | |
metacyclic 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 64 and has ID 2 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(64,2)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(64,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [64,2]
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 the DirectProduct and CyclicGroup functions as:
DirectProduct(CyclicGroup(8),CyclicGroup(8))