Dihedral group:D8 - Revision history

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

2024-12-03T00:36:51Z

Other properties

← Older revision		Revision as of 00:36, 3 December 2024
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	\|[[Satisfies property::Group isomorphic to its automorphism group]] \|\| Yes \|\| \|\|		\|[[Satisfies property::Group isomorphic to its automorphism group]] \|\| Yes \|\| \|\|
			\|-
			\|[[Satisfies property::2-Engel group]] \|\| Yes \|\| \|\|
	\|}		\|}

R-a-jones at 16:27, 12 January 2024

2024-01-12T16:27:57Z

← Older revision		Revision as of 16:27, 12 January 2024
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	{{particular group}}		{{particular group}}
	[[Importance rank::2\| ]]		[[Importance rank::2\| ]]
			[[Category:Dihedral groups]]
	==Definition==		==Definition==

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

2023-12-29T04:18:35Z

Other properties

← Older revision		Revision as of 04:18, 29 December 2023
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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 [[quaternion group]].		\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. \|\| Joint smallest non-A-group, alongside [[quaternion group]].
			\|-
			\|[[Satisfies property::Group isomorphic to its automorphism group]] \|\| Yes \|\| \|\|
	\|}		\|}

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

2023-12-14T16:56:31Z

Other properties

← Older revision		Revision as of 16:56, 14 December 2023
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	\|[[Dissatisfies property::Schur-trivial group]] \|\| No \|\| the [[Schur multiplier]] is [[cyclic group:Z2]]; see [[group cohomology of dihedral group:D8]]. \|\|		\|[[Dissatisfies property::Schur-trivial group]] \|\| No \|\| the [[Schur multiplier]] is [[cyclic group:Z2]]; see [[group cohomology of dihedral group:D8]]. \|\|
			\|-
			\| [[dissatisfies property::A-group]] \|\| No \|\| The only Sylow subgroup is itself, which is not abelian. \|\| Joint smallest non-A-group, alongside [[quaternion group]].
	\|}		\|}

R-a-jones: /* Linear representation theory */

2023-11-07T14:19:36Z

Linear representation theory

← Older revision		Revision as of 14:19, 7 November 2023
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	{{#lst:linear representation theory of dihedral group:D8\|character table}}		{{#lst:linear representation theory of dihedral group:D8\|character table}}


			==Galois theory==

			===Galois extensions===

			{{further\|[[Galois extensions for dihedral group:D8]]}}

	==Fusion systems==		==Fusion systems==

Xubia: /* Definition by presentation */ I added one accepted name for the group

2015-02-14T13:15:47Z

Definition by presentation: I added one accepted name for the group

← Older revision		Revision as of 13:15, 14 February 2015
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	===Definition by presentation===		===Definition by presentation===

	The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or the dihedral group of degree four (since its natural action is on four elements), is defined by the following [[presentation of a group\|presentation]], with <math>e</math> denoting the identity element:		The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or ''the dihedral group of degree four'' (since its natural action is on four elements), or sometimes the ''octic group'', is defined by the following [[presentation of a group\|presentation]], with <math>e</math> denoting the identity element:

	<math>\langle x,a \mid a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math>		<math>\langle x,a \mid a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math>

Vipul: /* Other properties */

2013-01-14T02:09:19Z

Other properties

← Older revision		Revision as of 02:09, 14 January 2013
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	\|[[Dissatisfies property::splitting-simple group]] \|\| No \|\| \|\|		\|[[Dissatisfies property::splitting-simple group]] \|\| No \|\| \|\|
			\|-
			\|[[Satisfies property::stem group]] \|\| Yes \|\| the center equals the derived subgroup, and hence, in particular, is contained in the derived subgroup. \|\|
			\|-
			\|[[Dissatisfies property::Schur-trivial group]] \|\| No \|\| the [[Schur multiplier]] is [[cyclic group:Z2]]; see [[group cohomology of dihedral group:D8]]. \|\|
	\|}		\|}

Vipul: /* GAP implementation */

2012-09-16T01:06:24Z

GAP implementation

← Older revision		Revision as of 01:06, 16 September 2012
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	==GAP implementation==		==GAP implementation==
			{{access GAP implementation online using SAGE\|id=5013}}
	{{GAP ID\|8\|3}}		{{GAP ID\|8\|3}}
	{{HallSenior\|8\|4}}		{{HallSenior\|8\|4}}

Vipul: /* Definition by presentation */

2012-09-16T00:56:21Z

Definition by presentation

← Older revision		Revision as of 00:56, 16 September 2012
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	===Definition by presentation===		===Definition by presentation===

	The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or the dihedral group of degree four (since its natural action is on four elements), is defined by the following [[presentation of a group\|presentation]]:		The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or the dihedral group of degree four (since its natural action is on four elements), is defined by the following [[presentation of a group\|presentation]], with <math>e</math> denoting the identity element:

	<math>\langle x,a \mid a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math>		<math>\langle x,a \mid a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math>
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	{{quotation\|Confused about presentations in general or this one in particular? If you're new to this stuff, check out [[constructing dihedral group:D8 from its presentation]]. Sophisticated group theorists need simply recall that [[presentation of semidirect product is disjoint union of presentations plus action by conjugation relations]]}}		{{quotation\|Confused about presentations in general or this one in particular? If you're new to this stuff, check out [[constructing dihedral group:D8 from its presentation]]. Sophisticated group theorists need simply recall that [[presentation of semidirect product is disjoint union of presentations plus action by conjugation relations]]}}

	===Geometric definition===		===Geometric definition===

Vipul: /* Other definitions */

2012-09-13T22:06:48Z

Other definitions

← Older revision		Revision as of 22:06, 13 September 2012
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	# The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S5\|symmetric group on five letters]].		# The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S5\|symmetric group on five letters]].
	# The <math>2</math>-[[Sylow subgroup]] of the [[alternating group:A6\|alternating group on six letters]].		# The <math>2</math>-[[Sylow subgroup]] of the [[alternating group:A6\|alternating group on six letters]].
	# The <math>2</math>-[[Sylow subgroup]] of [[projective special linear group:PSL(3,2)\|PSL(3,2)]].		# The [[member of family::unitriangular matrix group of degree three]] <math>UT(3,2)</math> over [[field:F2]], <math>2</math>-[[Sylow subgroup]] of [[projective special linear group:PSL(3,2)\|PSL(3,2)]].
	# The [[extraspecial group]] of order <math>2^3</math> and type '+'.		# The [[member of family::extraspecial group]] of order <math>2^3</math> and type '+'.

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