Quaternion group - Revision history

R-a-jones: /* Other properties */

2024-12-03T00:35:16Z

Other properties

← Older revision		Revision as of 00:35, 3 December 2024
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	\|-		\|-
	\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. \|\| Joint smallest non-A-group, alongside [[dihedral group:D8]].		\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. \|\| Joint smallest non-A-group, alongside [[dihedral group:D8]].
			\|-
			\|[[Satisfies property::2-Engel group]] \|\| Yes \|\| \|\|
	\|}		\|}

2024-01-13T19:46:29Z

← Older revision		Revision as of 19:46, 13 January 2024
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	This image shows visually the [[Cayley table]] of the quaternion group, with each element given a different colour. The elements are organized as in the above table: the first column/row corresponds to <math>1</math>, the second column/row corresponds to <math>-1</math>, etc.		This image shows visually the [[Cayley table]] of the quaternion group, with each element given a different colour. The elements are organized as in the above table: the first column/row corresponds to <math>1</math>, the second column/row corresponds to <math>-1</math>, etc.

	[[File:Quaternion group Cayley table visual.png\|~~400px~~]]		[[File:Quaternion group Cayley table visual.png\|200px]]

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

2024-01-13T19:46:15Z

Multiplication table

← Older revision		Revision as of 19:46, 13 January 2024
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	{{#lst:element structure of quaternion group\|multiplication table}}		{{#lst:element structure of quaternion group\|multiplication table}}

			This image shows visually the [[Cayley table]] of the quaternion group, with each element given a different colour. The elements are organized as in the above table: the first column/row corresponds to <math>1</math>, the second column/row corresponds to <math>-1</math>, etc.

			[[File:Quaternion group Cayley table visual.png\|400px]]

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

2023-12-26T01:05:27Z

Subgroups

← Older revision		Revision as of 01:05, 26 December 2023
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	==Subgroups==		==Subgroups==
	{{further\|[[Subgroup structure of quaternion group]]}}		{{further\|[[Subgroup structure of quaternion group]]}}

	[[Image:Q8latticeofsubgroups.png\|500px]]		[[Image:Q8latticeofsubgroups.png\|500px]]
	{{#lst:subgroup structure of quaternion group\|summary}}		{{#lst:subgroup structure of quaternion group\|summary}}

2023-12-15T15:49:37Z

← Older revision		Revision as of 15:49, 15 December 2023
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	{{TOCright}}		{{TOCright}}
	[[importance rank::2\| ]]		[[importance rank::2\| ]]
			[[Category:Generalized quaternion groups]]
	==Definition==		==Definition==

2023-12-14T16:56:02Z

Other properties

← Older revision		Revision as of 16:56, 14 December 2023
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	\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|		\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|
	\|-		\|-
	\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. Joint smallest non-A-group, alongside [[dihedral group:D8]]. \|\|		\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. \|\| Joint smallest non-A-group, alongside [[dihedral group:D8]].
	\|}		\|}

2023-12-14T16:55:48Z

Other properties

← Older revision		Revision as of 16:55, 14 December 2023
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	\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|		\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|
	\|-		\|-
	\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. Joint smallest non-A-group, alongside [[dihedral group:D8]].		\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. Joint smallest non-A-group, alongside [[dihedral group:D8]]. \|\|
	\|}		\|}

2023-12-14T16:55:38Z

Other properties

← Older revision		Revision as of 16:55, 14 December 2023
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	\|-		\|-
	\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|		\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|
			\|-
			\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. Joint smallest non-A-group, alongside [[dihedral group:D8]].
	\|}		\|}

2013-01-10T03:36:55Z

Other properties

← Older revision		Revision as of 03:36, 10 January 2013
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	===Other properties===		===Other properties===

	~~<toggledisplay>~~
	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	!Property !! Satisfied !! Explanation !! Comment		!Property !! Satisfied !! Explanation !! Comment
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	\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|		\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|
	\|}		\|}
	~~</toggledisplay>~~

	==Subgroups==		==Subgroups==

2013-01-10T03:32:44Z

Other properties

← Older revision		Revision as of 03:32, 10 January 2013
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	\|-		\|-
	\|[[Satisfies property::splitting-simple group]] \|\| Yes \|\| \|\|		\|[[Satisfies property::splitting-simple group]] \|\| Yes \|\| \|\|
			\|-
			\|[[Satisfies property::Schur-trivial group]] \|\| Yes \|\| See [[group cohomology of quaternion group]] \|\|
	\|}		\|}
	</toggledisplay>		</toggledisplay>