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

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	8	groups with same order
prime-base logarithm of order	3	groups with same prime-base logarithm of order
max-length of a group	3		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	3		chief length equals prime-base logarithm of order for group of prime power order
composition length	3		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	2	groups with same order and exponent of a group | groups with same prime-base logarithm of order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	1	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length	1	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length	Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set	3	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	3	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group	same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group	3	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group	3	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group	3	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class	1		The group is a nontrivial abelian group
derived length	1		The group is a nontrivial abelian group
Fitting length	1		The group is a nontrivial abelian group

Group properties

Property	Satisfied?	Explanation
abelian group	Yes
elementary abelian group	Yes
cyclic group	No
metacyclic group	No
homocyclic group	Yes
group of prime power order	Yes
nilpotent group	Yes
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 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)