Closure-characteristic subgroup - Revision history

Vipul: /* Stronger properties */

2014-12-21T17:30:00Z

Stronger properties

← Older revision		Revision as of 17:30, 21 December 2014
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	\| [[Weaker than::characteristic subgroup]] \|\| invariant under all [[automorphism]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|characteristic subgroup}}		\| [[Weaker than::characteristic subgroup]] \|\| invariant under all [[automorphism]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|characteristic subgroup}}
	\|-		\|-
	\| [[Weaker than::automorph-conjugate subgroup]] \|\| all [[automorphic subgroups]] are [[conjugate subgroups\|conjugate]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-conjugate subgroup}}		\| [[Weaker than::automorph-dominating subgroup]] \|\| all [[automorphic subgroups]] are contained in [[conjugate subgroups]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-dominating subgroup}}
			\|-
			\| [[Weaker than::automorph-conjugate subgroup]] \|\| all [[automorphic subgroups]] are [[conjugate subgroups\|conjugate]] \|\| (via automorph-dominating) \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-conjugate subgroup}}
	\|-		\|-
	\| [[Weaker than::join of automorph-conjugate subgroups]] \|\| [[join of subgroups\|join]] of [[automorph-conjugate subgroup]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}		\| [[Weaker than::join of automorph-conjugate subgroups]] \|\| [[join of subgroups\|join]] of [[automorph-conjugate subgroup]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}

Vipul: /* Stronger properties */

2014-12-20T21:26:53Z

Stronger properties

← Older revision		Revision as of 21:26, 20 December 2014
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::~~Characteristic~~ subgroup]] \|\| invariant under all [[automorphism]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|characteristic subgroup}}		\| [[Weaker than::characteristic subgroup]] \|\| invariant under all [[automorphism]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|characteristic subgroup}}
	\|-		\|-
	\| [[Weaker than::~~Automorph~~-conjugate subgroup]] \|\| all [[automorphic subgroups]] are [[conjugate subgroups\|conjugate]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-conjugate subgroup}}		\| [[Weaker than::automorph-conjugate subgroup]] \|\| all [[automorphic subgroups]] are [[conjugate subgroups\|conjugate]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-conjugate subgroup}}
	\|-		\|-
	\| [[Weaker than::~~Join~~ of automorph-conjugate subgroups]] \|\| [[join of subgroups\|join]] of [[automorph-conjugate subgroup]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}		\| [[Weaker than::join of automorph-conjugate subgroups]] \|\| [[join of subgroups\|join]] of [[automorph-conjugate subgroup]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}
	\|-		\|-
	\| [[Weaker than::Sylow subgroup]] \|\| <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Sylow subgroup}}		\| [[Weaker than::Sylow subgroup]] \|\| <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Sylow subgroup}}
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	\| [[Weaker than::Hall subgroup]] \|\| subgroup of finite group whose order and index are relatively prime \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}		\| [[Weaker than::Hall subgroup]] \|\| subgroup of finite group whose order and index are relatively prime \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}
	\|-		\|-
	\| [[Weaker than::~~Contranormal~~ subgroup]] \|\| [[normal closure]] is whole group \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|contranormal subgroup}}		\| [[Weaker than::contranormal subgroup]] \|\| [[normal closure]] is whole group \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|contranormal subgroup}}
	\|}		\|}

Vipul: /* Metaproperties */

2014-12-20T21:26:29Z

Metaproperties

← Older revision		Revision as of 21:26, 20 December 2014
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	{{wikilocal-section}}		{{wikilocal-section}}

	~~Here is a summary:~~

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Metaproperties */

2014-12-20T21:26:14Z

Metaproperties

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	==Metaproperties==		==Metaproperties==

	{{~~trim~~}}		{{wikilocal-section}}

	{{join-closed}}		Here is a summary:

			{\| class="sortable" border="1"
			!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
			\|-
			\| [[satisfies metaproperty::trim subgroup property]] \|\| Yes \|\| (obvious) \|\| 0 \|\| In any group <math>G</math>, both the trivial subgroup <math>\{ e \}</math> and the whole group <math>G</math> are closure-characteristic.
			\|-
			\| [[satisfies metaproperty::strongly join-closed subgroup property]] \|\| Yes \|\| [[closure-characteristicity is strongly join-closed]] \|\| {{#show: closure-characteristicity is strongly join-closed\| ?Difficulty level}} \|\| Suppose <math>G</math> is a group and <math>H_i, i \in I</math> are all closure-characteristic subgroups of <math>G</math>. Then the join <math>\langle H_i \rangle_{i \in I}</math> is also closure-characteristic.
			\|}

Vipul: /* Stronger properties */

2014-12-20T19:14:35Z

Stronger properties

← Older revision		Revision as of 19:14, 20 December 2014
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	\| [[Weaker than::Automorph-conjugate subgroup]] \|\| all [[automorphic subgroups]] are [[conjugate subgroups\|conjugate]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-conjugate subgroup}}		\| [[Weaker than::Automorph-conjugate subgroup]] \|\| all [[automorphic subgroups]] are [[conjugate subgroups\|conjugate]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-conjugate subgroup}}
	\|-		\|-
	\| [[Weaker than::Join of automorph-conjugate subgroups]] \|\| [[join of subgroups\|join]] of [[automorph-conjugate ~~subgroups~~]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}		\| [[Weaker than::Join of automorph-conjugate subgroups]] \|\| [[join of subgroups\|join]] of [[automorph-conjugate subgroup]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}
	\|-		\|-
	\| [[Weaker than::Sylow subgroup]] \|\| <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Sylow subgroup}}		\| [[Weaker than::Sylow subgroup]] \|\| <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Sylow subgroup}}
	\|-		\|-
	\| [[Weaker than::join of Sylow subgroups]] \|\| join of Sylow ~~subgroups~~ \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of Sylow subgroups}}		\| [[Weaker than::join of Sylow subgroups]] \|\| join of [[Sylow subgroup]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of Sylow subgroups}}
	\|-		\|-
	\| [[Weaker than::Hall subgroup]] \|\| subgroup of finite group whose order and index are relatively prime \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}		\| [[Weaker than::Hall subgroup]] \|\| subgroup of finite group whose order and index are relatively prime \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}

Vipul: /* Relation with other properties */

2014-12-20T19:14:12Z

Relation with other properties

← Older revision		Revision as of 19:14, 20 December 2014
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	\|-		\|-
	\| [[Weaker than::Sylow subgroup]] \|\| <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Sylow subgroup}}		\| [[Weaker than::Sylow subgroup]] \|\| <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Sylow subgroup}}
			\|-
			\| [[Weaker than::join of Sylow subgroups]] \|\| join of Sylow subgroups \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of Sylow subgroups}}
	\|-		\|-
	\| [[Weaker than::Hall subgroup]] \|\| subgroup of finite group whose order and index are relatively prime \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}		\| [[Weaker than::Hall subgroup]] \|\| subgroup of finite group whose order and index are relatively prime \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}

Vipul: /* Definition */

2014-12-20T19:12:33Z

Definition

← Older revision		Revision as of 19:12, 20 December 2014
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	{{quick phrase\|[[quick phrase::normal closure is characteristic]], [[quick phrase::join of all conjugates is characteristic]]}}		{{quick phrase\|[[quick phrase::normal closure is characteristic]], [[quick phrase::join of all conjugates is characteristic]]}}
	~~===Symbol-free definition===~~

	A [[subgroup]] of a [[group]] is termed '''closure-characteristic''' if its [[defining ingredient::normal closure]] in the whole group is a [[defining ingredient::characteristic subgroup]].		A [[subgroup]] of a [[group]] is termed '''closure-characteristic''', or a '''subgroup whose normal closure is characteristic''', if its [[defining ingredient::normal closure]] in the whole group is a [[defining ingredient::characteristic subgroup]]. In symbols, a [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''closure-characteristic''' if the [[normal closure]] <math>H^G</math> of <math>H</math> in <math>G</math> is [[characteristic subgroup\|characteristic]] in <math>G</math>.

	~~===Definition with~~ symbols~~===~~

	A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''closure-characteristic''' if the [[normal closure]] <math>H^G</math> of <math>H</math> in <math>G</math> is [[characteristic subgroup\|characteristic]] in <math>G</math>.

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Relation with other properties */

2010-01-11T23:47:20Z

Relation with other properties

← Older revision		Revision as of 23:47, 11 January 2010
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	===Stronger properties===		===Stronger properties===

	* [[Characteristic subgroup]]		{\| class="sortable" border="1"
	* [[Automorph-conjugate subgroup]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Join of automorph-conjugate subgroups]]		\|-
	* [[Sylow subgroup]]		\| [[Weaker than::Characteristic subgroup]] \|\| invariant under all [[automorphism]]s \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|characteristic subgroup}}
	* [[Hall subgroup]]		\|-
	* [[Contranormal subgroup]]		\| [[Weaker than::Automorph-conjugate subgroup]] \|\| all [[automorphic subgroups]] are [[conjugate subgroups\|conjugate]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|automorph-conjugate subgroup}}
			\|-
			\| [[Weaker than::Join of automorph-conjugate subgroups]] \|\| [[join of subgroups\|join]] of [[automorph-conjugate subgroups]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}
			\|-
			\| [[Weaker than::Sylow subgroup]] \|\| <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Sylow subgroup}}
			\|-
			\| [[Weaker than::Hall subgroup]] \|\| subgroup of finite group whose order and index are relatively prime \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}
			\|-
			\| [[Weaker than::Contranormal subgroup]] \|\| [[normal closure]] is whole group \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|contranormal subgroup}}
			\|}

	===Conjunction with other properties===		===Conjunction with other properties===

Vipul at 23:33, 11 January 2010

2010-01-11T23:33:46Z

← Older revision		Revision as of 23:33, 11 January 2010
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	==Definition==		==Definition==

			{{quick phrase\|[[quick phrase::normal closure is characteristic]], [[quick phrase::join of all conjugates is characteristic]]}}
	===Symbol-free definition===		===Symbol-free definition===

	A [[subgroup]] of a [[group]] is termed '''closure-characteristic''' if its [[normal closure]] in the whole group is a [[characteristic subgroup]].		A [[subgroup]] of a [[group]] is termed '''closure-characteristic''' if its [[defining ingredient::normal closure]] in the whole group is a [[defining ingredient::characteristic subgroup]].

	===Definition with symbols===		===Definition with symbols===

Vipul: 4 revisions

2008-05-07T23:18:24Z

4 revisions

← Older revision	Revision as of 23:18, 7 May 2008
(No difference)