# Direct product of cyclic group of prime-cube order and cyclic group of prime 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

This group is defined as the external direct product of the cyclic group of prime-cube order (denoted or ) and the cyclic group of prime order (denoted or ), i.e., it is defined as (or, in alternative notation, ). It corresponds to the partition (see classification of finitely generated abelian groups and abelian group of prime power order):

For a given prime , it can also be defined as the unique (up to isomorphism) abelian group of order and exponent .

## Particular cases

Value of prime number | Corresponding group |
---|---|

2 | direct product of Z8 and Z2 |

3 | direct product of Z27 and Z3 |

5 | direct product of Z125 and Z5 |

## Arithmetic functions

## GAP implementation

### Group ID

This finite group has order p^4 and has ID 5 among the group of order p^4 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(p^4,5)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(p^4,5);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [p^4,5]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Short descriptions

Description | Functions used | Mathematical comments |
---|---|---|

DirectProduct(CyclicGroup(p^3),CyclicGroup(p)) |
DirectProduct and CyclicGroup |