# Elementary abelian group:E16

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## Contents

## Definition

The **elementary abelian group of order sixteen** is defined in the following equivalent ways:

- It is the elementary abelian group of order sixteen.
- It is the additive group of a four-dimensional vector space over the field of two elements.
- It is the additive group of the field of sixteen elements.
- It is the direct product of four copies of cyclic group:Z2.
- It is the direct product of two copies of the Klein four-group.
- It is the Burnside group : the
*free group*of rank four and exponent two.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 16 and has ID 14 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,14)`

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

`gap> G := SmallGroup(16,14);`

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

`IdGroup(G) = [16,14]`

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 also be described using GAP's ElementaryAbelianGroup function:

ElementaryAbelianGroup(16)