# Elementary abelian 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 **elementary abelian group of order** , denoted , is the elementary abelian group whose order is . In other words, it is (up to isomorphism) the external direct product of three copies of the group of prime order.

## Particular cases

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

2 | elementary abelian group:E8 |

3 | elementary abelian group:E27 |

5 | elementary abelian group:E125 |

7 | elementary abelian group:E343 |

## Arithmetic functions

## GAP implementation

### Group ID

This finite group has order p^3 and has ID 5 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^3,5)`

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

`gap> G := SmallGroup(p^3,5);`

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

`IdGroup(G) = [p^3,5]`

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 also be defined using GAP's ElementaryAbelianGroup function:

`ElementaryAbelianGroup(p^3)`