# 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.