# Elementary abelian group:E8

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## Contents

## Definition

The **elementary abelian group of order eight** is defined as followed:

- It is the elementary abelian group of order eight.
- It is the additive group of a three-dimensional vector space over a field of two elements.
- It is the only abelian group of order eight and exponent two.
- It is the generalized dihedral group corresponding to the Klein four-group.
- It is the Burnside group : the
*free group*of rank three and exponent two.

## Position in classifications

Type of classification | Name in that classification |
---|---|

GAP ID | (8,5), i.e., 5th among the groups of order 8 |

Hall-Senior number | 1 among groups of order 8 |

Hall-Senior symbol |

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions

## Group properties

Property | Satisfied? | Corollary properties satisfied/dissatisfied |
---|---|---|

elementary abelian group | Yes | Satisfies: abelian group, nilpotent group, group of prime power order, homocyclic group |

cyclic group | No | |

metacyclic group | No | |

rational group | Yes | |

rational-representation group | Yes |

## GAP implementation

### Group ID

This finite group has order 8 and has ID 5 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(8,5)`

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

`gap> G := SmallGroup(8,5);`

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

`IdGroup(G) = [8,5]`

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(8)`