Elementary abelian group:E8

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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.
• Is is the direct product $\mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}$.

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?	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

Automorphism group

The automorphism group of this group is PSL(2, 7), the second smallest non-cyclic simple group.

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)