Elementary abelian group:E8
From Groupprops
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]
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)