Elementary abelian group:E8

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Definition

The elementary abelian group of order eight is defined as followed:

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

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 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)