Elementary abelian group:E8 - Revision history

R-a-jones: automorphism group info

2023-06-05T15:55:01Z

automorphism group info

← Older revision		Revision as of 15:55, 5 June 2023
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	\| [[satisfies property::rational-representation group]] \|\| Yes \|\|		\| [[satisfies property::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==		==GAP implementation==

2023-06-05T15:53:52Z

added definition that newer students of group theory may be familar with

← Older revision		Revision as of 15:53, 5 June 2023
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	* It is the [[member of family::generalized dihedral group]] corresponding to the [[Klein four-group]].		* It is the [[member of family::generalized dihedral group]] corresponding to the [[Klein four-group]].
	* It is the [[member of family::Burnside group]] <math>B(3,2)</math>: the ''free group'' of rank three and exponent two.		* It is the [[member of family::Burnside group]] <math>B(3,2)</math>: the ''free group'' of rank three and exponent two.
			* Is is the direct product <math>\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2</math>.

	==Position in classifications==		==Position in classifications==

2011-05-19T23:52:40Z

Definition

← Older revision		Revision as of 23:52, 19 May 2011
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	The '''elementary abelian group of order eight''' is defined as followed:		The '''elementary abelian group of order eight''' is defined as followed:

	* It is the [[elementary abelian group]] of order eight.		* It is the [[defining ingredient::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 additive group of a three-dimensional vector space over a [[defining ingredient::field:F2\|field of two elements]].
	* It is the only abelian group of order eight and exponent two.		* It is the only abelian group of order eight and exponent two.
	* It is the [[member of family::generalized dihedral group]] corresponding to the [[Klein four-group]].		* It is the [[member of family::generalized dihedral group]] corresponding to the [[Klein four-group]].

2011-05-19T23:51:53Z

Group properties

← Older revision		Revision as of 23:51, 19 May 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Property !! Satisfied? !! ~~Explanation~~		! Property !! Satisfied? !! Corollary properties satisfied/dissatisfied
	\|-		\|-
	\| [[satisfies property::abelian group]] \|\| Yes \|\|		\| [[satisfies property::elementary abelian group]] \|\| Yes \|\| Satisfies: [[satisfies property::abelian group]], [[satisfies property::nilpotent group]], [[satisfies property::group of prime power order]], [[satisfies property::homocyclic group]]
	\|-
	\| [[satisfies property::~~elementary~~ abelian group]] ~~\|\| Yes \|\|~~
	\|-		\|-
	\| [[dissatisfies property::cyclic group]] \|\| No \|\|		\| [[dissatisfies property::cyclic group]] \|\| No \|\|
	\|-		\|-
	\| [[dissatisfies property::metacyclic group]] \|\| No \|\|		\| [[dissatisfies property::metacyclic group]] \|\| No \|\|
	\|-
	~~\| [[satisfies property::homocyclic group]] \|\| Yes \|\|~~
	\|-
	~~\| [[satisfies property::group of prime power order]] \|\| Yes \|\|~~
	\|-
	~~\| [[satisfies property::nilpotent group]] \|\| Yes \|\|~~
	\|-		\|-
	\| [[satisfies property::rational group]] \|\| Yes \|\|		\| [[satisfies property::rational group]] \|\| Yes \|\|
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	\| [[satisfies property::rational-representation group]] \|\| Yes \|\|		\| [[satisfies property::rational-representation group]] \|\| Yes \|\|
	\|}		\|}

	==GAP implementation==		==GAP implementation==

2011-05-19T23:50:02Z

← Older revision		Revision as of 23:50, 19 May 2011
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	number of subgroups = 16}}		number of subgroups = 16}}

			==Group properties==

			{\| class="sortable" border="1"
			! Property !! Satisfied? !! Explanation
			\|-
			\| [[satisfies property::abelian group]] \|\| Yes \|\|
			\|-
			\| [[satisfies property::elementary abelian group]] \|\| Yes \|\|
			\|-
			\| [[dissatisfies property::cyclic group]] \|\| No \|\|
			\|-
			\| [[dissatisfies property::metacyclic group]] \|\| No \|\|
			\|-
			\| [[satisfies property::homocyclic group]] \|\| Yes \|\|
			\|-
			\| [[satisfies property::group of prime power order]] \|\| Yes \|\|
			\|-
			\| [[satisfies property::nilpotent group]] \|\| Yes \|\|
			\|-
			\| [[satisfies property::rational group]] \|\| Yes \|\|
			\|-
			\| [[satisfies property::rational-representation group]] \|\| Yes \|\|
			\|}
	==GAP implementation==		==GAP implementation==

2010-07-04T02:05:28Z

Arithmetic functions

2010-07-04T02:00:55Z

Arithmetic functions

@@ 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}}
 ==GAP implementation==

2010-03-26T21:21:19Z

← Older revision		Revision as of 21:21, 26 March 2010
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	* It is the [[member of family::Burnside group]] <math>B(3,2)</math>: the ''free group'' of rank three and exponent two.		* It is the [[member of family::Burnside group]] <math>B(3,2)</math>: the ''free group'' of rank three and exponent two.

			==Position in classifications==

			{\| class="wikitable" border="1"
			! 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 \|\| <math>(1^3)</math>
			\|}
	==Arithmetic functions==		==Arithmetic functions==

2010-03-26T03:02:40Z

← Older revision		Revision as of 03:02, 26 March 2010
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	{\| class="wikitable" border="1"		{\| class="wikitable" border="1"
	! Function !! Value !! Explanation		! Function !! Value !! Explanation
			\|-
			\| [[underlying prime of p-group]] \|\| [[arithmetic function value::underlying prime of p-group;2\|2]]\|\|
	\|-		\|-
	\| [[Order of a group\|order]] \|\| [[arithmetic function value::order of a group;8\|8]] \|\|		\| [[Order of a group\|order]] \|\| [[arithmetic function value::order of a group;8\|8]] \|\|
			\|-
			\| [[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]] \|\|		\| [[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]] \|\|		\| [[nilpotency class]] \|\| [[arithmetic function value::nilpotency class;1\|1]] \|\|

2009-09-10T21:49:38Z

← Older revision	Revision as of 21:49, 10 September 2009
(No difference)