# Elementary abelian group:E81

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This group is the elementary abelian group of order . In other words, it is the additive group of a four-dimensional vector space over the field of three elements. Equivalently, it is the direct product of four copies of the cyclic group of order three.

## Contents

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 81#Arithmetic functions

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | Yes | |

elementary abelian group | Yes | |

group of prime power order | Yes | |

homocyclic group | Yes | |

characteristically simple group | Yes |

## GAP implementation

### Group ID

This finite group has order 81 and has ID 15 among the groups of order 81 in GAP's SmallGroup library. For context, there are 15 groups of order 81. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(81,15)`

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

`gap> G := SmallGroup(81,15);`

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

`IdGroup(G) = [81,15]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can be defined using GAP's ElementaryAbelianGroup function:

`ElementaryAbelianGroup(81)`