Cyclic group of prime-cube order
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. The cyclic group of order
is a cyclic group (specifically, a finite cyclic group whose order is
. It can be defined by the presentation:
where is the identity element. The group is written
or
.
Particular cases
Value of prime number ![]() |
Corresponding group |
---|---|
2 | cyclic group:Z8 |
3 | cyclic group:Z27 |
5 | cyclic group:Z125 |
Arithmetic functions
GAP implementation
Group ID
This finite group has order p^3 and has ID 1 among the group of order p^3 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(p^3,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(p^3,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [p^3,1]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Alternative descriptions
The group can be described using GAP's CyclicGroup function:
CyclicGroup(p^3)
where we can replace by a particular prime or replace
by a particular cube of a prime.