Direct product of Z4 and Z4
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This group, denoted or is defined in the following equivalent ways:
- It is a homocyclic group of order sixteen and exponent four.
- It is the direct product of two copies of cyclic group:Z4.
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-square order and cyclic group of prime-square order|
|3||direct product of Z9 and Z9|
|5||direct product of Z25 and Z25|
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions
Want to compare and contrast group properties with other groups of the same order? Check out groups of order 16#Group properties
|Abelian group||Yes||Direct product of cyclic groups|
|Nilpotent group||Yes||Abelian implies nilpotent|
This finite group has order 16 and has ID 2 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. 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 :
gap> G := SmallGroup(16,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [16,2]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
The group can also be defined using GAP's DirectProduct function: