# Elementary abelian group:E27

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## Contents

## Definition

This group, sometimes denoted , is the elementary abelian group of order . In other words, it is the additive group of a three-dimensional vector space over the field of three elements. Equivalently, it is the additive group of the field of 27 elements. Equivalently, it is the direct product of three copies of the cyclic group of order three.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 27#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 27 and has ID 5 among the groups of order 27 in GAP's SmallGroup library. For context, there are groups of order 27. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(27,5)`

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

`gap> G := SmallGroup(27,5);`

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

`IdGroup(G) = [27,5]`

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(27)`