Elementary abelian group:E8: Difference between revisions

Revision as of 02:00, 4 July 2010

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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 -

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)

@@ Line 24: / Line 24: @@
 ==Arithmetic functions==
-{| class="wikitable" border="1"
+{{abelian p-group arithmetic function table|
-! Function !! Value !! Explanation
+underlying prime = 2|
-|-
+order = 8|
-| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]]||
+order p-log = 3|
-|-
+exponent = 2|
-| [[Order of a group|order]] || [[arithmetic function value::order of a group;8|8]] ||
+exponent p-log = 1|
-|-
+rank = 3}}
-| [[prime-base logarithm of order]] || [[arithmetic function value::prime-base logarithm of order;3|3]] ||
-|-
-| [[Exponent of a group|exponent]] || [[arithmetic function value::exponent of a group;2|2]] ||
-|-
-| [[prime-base logarithm of exponent]] || [[arithmetic function value::prime-base logarithm of exponent;1|1]] ||
-|-
-| [[nilpotency class]] || [[arithmetic function value::nilpotency class;1|1]] ||
-|-
-| [[derived length]] || [[arithmetic function value::derived length;1|1]] ||
-|-
-| [[subgroup rank of a group|subgroup rank]] || [[arithmetic function value::subgroup rank of a group;3|3]] ||
-|-
-| [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;3|3]] ||
-|-
-| [[max-length of a group|max-length]] || [[arithmetic function value::max-length of a group;3|3]] ||
-|-
-| [[rank of a p-group|rank]] || [[arithmetic function value::rank of a p-group;3|3]] ||
-|-
-| [[normal rank of a p-group|normal rank]] || [[arithmetic function value::normal rank of a p-group;3|3]] ||
-|-
-| [[characteristic rank of a p-group|characteristic rank]]|| [[arithmetic function value::characteristic rank of a p-group;3|3]] ||
-|-
-| [[number of subgroups]] || [[arithmetic function value::number of subgroups;16|16]] ||
-|-
-| [[number of conjugacy classes]] || [[arithmetic function value::number of conjugacy classes;8|8]] ||
-|}
 ==GAP implementation==