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

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.