Direct product of cyclic group of prime-square order and cyclic group of prime-square order
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This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups
Contents
Definition
Let be a prime number. This group is defined in the following equivalent ways:
- It is the external direct product of two copies of the cyclic group of prime-square order, i.e., it is the group
.
- It is the homocyclic group of order
and exponent
.
Particular cases
Value of prime number ![]() |
Corresponding group |
---|---|
2 | direct product of Z4 and Z4 |
3 | direct product of Z9 and Z9 |
5 | direct product of Z25 and Z25 |
Arithmetic functions
GAP implementation
Group ID
This finite group has order p^4 and has ID 2 among the groups of order p^4 in GAP's SmallGroup library. For context, there are groups of order p^4. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(p^4,2)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(p^4,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [p^4,2]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Short descriptions
Description | Functions used | Mathematical comments |
---|---|---|
DirectProduct(CyclicGroup(p^2),CyclicGroup(p^2)) | DirectProduct and CyclicGroup |