Elementary abelian group:E8

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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 $B(3,2)$: 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	$(1^{3})$

Arithmetic functions

Function	Value	Explanation
underlying prime of p-group	2
order	8
prime-base logarithm of order	3
exponent	2
prime-base logarithm of exponent	1
nilpotency class	1
derived length	1
subgroup rank	3
minimum size of generating set	3
max-length	3
rank	3
normal rank	3
characteristic rank	3
number of subgroups	16
number of conjugacy classes	8

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 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 $G$:

gap> G := SmallGroup(8,5);

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