# Elementary abelian group:E16

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## 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 $B(4,2)$: the free group of rank four and exponent two.

## Arithmetic functions

Function Value Explanation
order 16
exponent 2
Frattini length 1
max-length 4
minimum size of generating set 4
rank as p-group 4
normal rank 4
characteristic rank 4
subgroup rank 4
number of subgroups 67
number of conjugacy classes 16
number of conjugacy classes of subgroups 67

## 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 $G$:

gap> G := SmallGroup(16,14);

Conversely, to check whether a given group $G$ 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)`