# Cyclic group of prime-cube order

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.