Elementary abelian group:E8
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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|
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions
|Property||Satisfied?||Corollary properties satisfied/dissatisfied|
|elementary abelian group||Yes||Satisfies: abelian group, nilpotent group, group of prime power order, homocyclic group|
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:
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:
to have GAP output the group ID, that we can then compare to what we want.
The group can be defined using GAP's ElementaryAbelianGroup function: