# Elementary abelian group:E9

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## Contents

## Definition

This elementary abelian group is defined in the following equivalent ways:

- It is the elementary abelian group of prime-square order where the prime is three.
- It is the additive group of the field of nine elements.
- It is a direct product of two 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 9#Arithmetic functions

## Group properties

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

cyclic group | No | |

elementary abelian group | Yes | |

metacyclic group | Yes | |

abelian group | Yes | |

nilpotent group | Yes | |

solvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 9 and has ID 2 among the groups of order 9 in GAP's SmallGroup library. For context, there are groups of order 9. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(9,2)`

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

`gap> G := SmallGroup(9,2);`

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

`IdGroup(G) = [9,2]`

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(9)`

It can also be defined using GAP's DirectProduct and CyclicGroup functions:

`DirectProduct(CyclicGroup(3),CyclicGroup(3))`