Cyclic group of prime-cube 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

Definition

Let p be a prime number. The cyclic group of order p^3 is a cyclic group (specifically, a finite cyclic group whose order is p^3. It can be defined by the presentation:

\langle a \mid a^{p^3} = e \rangle

where e is the identity element. The group is written \mathbb{Z}_{p^3}, C_{p^3} or \mathbb{Z}/p^3\mathbb{Z}.

Particular cases

Value of prime number p Corresponding group
2 cyclic group:Z8
3 cyclic group:Z27
5 cyclic group:Z125

Arithmetic functions

Function Value Similar groups Explanation for function value
prime-base logarithm of order 3 groups with same prime-base logarithm of order
max-length of a group 3 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 3 chief length equals prime-base logarithm of order for group of prime power order
composition length 3 composition length equals prime-base logarithm of order for group of prime power order
prime-base logarithm of exponent 3 groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length 3 groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set 1 groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 1 groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group 1 groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group 1 groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group 1 groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class 1 The group is a nontrivial abelian group
derived length 1 The group is a nontrivial abelian group
Fitting length 1 The group is a nontrivial abelian group

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 G:

gap> G := SmallGroup(p^3,1);

Conversely, to check whether a given group G 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 p by a particular prime or replace p^3 by a particular cube of a prime.