Join of automorph-conjugate subgroups - Revision history

Vipul: /* Weaker properties */

2014-12-20T21:17:48Z

Weaker properties

← Older revision		Revision as of 21:17, 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
	\|-		\|-
	\| [[Stronger than::closure-characteristic subgroup]] \|\| its [[normal closure]] is a [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}		\| [[Stronger than::closure-characteristic subgroup]] \|\| its [[normal closure]] is a [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}
			\|-
			\| [[Stronger than::normal-to-characteristic subgroup]] \|\| if normal, it is also characteristic \|\| \|\| \|\| {{intermediate notions short\|normal-to-characteristic subgroup\|join of automorph-conjugate subgroups}}
	\|}		\|}

Vipul: /* Relation with other properties */

2014-12-20T21:16:54Z

Relation with other properties

← Older revision		Revision as of 21:16, 20 December 2014
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	===Stronger properties===		===Stronger properties===

	* [[~~Automorph~~-conjugate subgroup]]		{\| class="sortable" border="1"
	* [[Hall subgroup]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::automorph-conjugate subgroup]] \|\| [[conjugate subgroups\|conjugate]] to all its [[automorphic subgroups]] \|\| (obvious) \|\| [[automorph-conjugacy is not finite-join-closed]] \|\| {{intermediate notions short\|join of automorph-conjugate subgroups\|automorph-conjugate subgroup}}
			\|-
			\| [[Weaker than::join of Sylow subgroups]] \|\| join of [[Sylow subgroup]]s \|\| (obvious) \|\| Any proper nontrivial characteristic subgroup of a <math>p</math>-group, e.g., [[Z2 in Z4]] \|\| {{intermediate notions short\|join of automorph-conjugate subgroups\|join of Sylow subgroups}}
			\|-
			\| [[Weaker than::Hall subgroup]] \|\| subgroup whose order and index are relatively prime \|\| ([[Hall implies join of Sylow subgroups\|via join of Sylow subgroups]]) \|\| (via join of Sylow subgroups) \|\| {{intermediate notions short\|join of automorph-conjugate subgroups\|Hall subgroups}}
			\|}

	===Weaker properties===		===Weaker properties===

	* [[~~Closure~~-characteristic subgroup]]		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::closure-characteristic subgroup]] \|\| its [[normal closure]] is a [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|join of automorph-conjugate subgroups}}
			\|}

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

	Any normal subgroup that is a join of automorph-conjugate subgroups is [[characteristic subgroup\|characteristic]]. {{normal-to-characteristic}}		Any normal subgroup that is a join of automorph-conjugate subgroups is [[characteristic subgroup\|characteristic]]. {{normal-to-characteristic}}

Vipul: 5 revisions

2008-05-07T23:45:54Z

5 revisions

← Older revision	Revision as of 23:45, 7 May 2008
(No difference)

Vipul: /* Conjunction with other properties */

2008-02-08T18:26:30Z

Conjunction with other properties

← Older revision		Revision as of 18:26, 8 February 2008
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	===Conjunction with other properties===		===Conjunction with other properties===

	Any normal subgroup that is a join of automorph-conjugate subgroups is [[characteristic subgroup\|characteristic]]. ~~Thus, being AC-generated is a [[~~normal-to-characteristic ~~subgroup property]].~~		Any normal subgroup that is a join of automorph-conjugate subgroups is [[characteristic subgroup\|characteristic]]. {{normal-to-characteristic}}

Vipul at 18:22, 7 February 2008

2008-02-07T18:22:19Z

← Older revision		Revision as of 18:22, 7 February 2008
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	==Definition==		==Definition==

	A [[subgroup]] of a [[group]] is ~~said to be~~ '''AC-generated''' if it is generated by a collection of [[automorph-conjugate subgroup]]s, viz it is a join of automorph-conjugate subgroups.		A [[subgroup]] of a [[group]] is termed a '''join of automorph-conjugate subgroups''' or '''AC-generated''' if it is generated by a collection of [[automorph-conjugate subgroup]]s, viz it is a join of automorph-conjugate subgroups.

	==~~Facts~~==		==Relation with other properties==

	Any normal AC-~~generated subgroup~~ is [[characteristic subgroup\|characteristic]]. Thus, being AC-generated is a [[normal-to-characteristic subgroup property]].		===Stronger properties===

			* [[Automorph-conjugate subgroup]]
			* [[Hall subgroup]]

			===Weaker properties===

			* [[Closure-characteristic subgroup]]

			===Conjunction with other properties===

			Any normal subgroup that is a join of automorph-conjugate subgroups is [[characteristic subgroup\|characteristic]]. Thus, being AC-generated is a [[normal-to-characteristic subgroup property]].

Vipul: AC-generated subgroup moved to Join of automorph-conjugate subgroups

2008-02-07T18:20:25Z

AC-generated subgroup moved to Join of automorph-conjugate subgroups

← Older revision	Revision as of 18:20, 7 February 2008
(No difference)

Vipul at 23:34, 22 September 2007

2007-09-22T23:34:33Z

← Older revision		Revision as of 23:34, 22 September 2007
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	{subgroup property}}		{{subgroup property}}

	{{wikilocal}}		{{wikilocal}}

Vipul at 23:34, 22 September 2007

2007-09-22T23:34:21Z

New page

{subgroup property}}

{{wikilocal}}

==Definition==

A [[subgroup]] of a [[group]] is said to be '''AC-generated''' if it is generated by a collection of [[automorph-conjugate subgroup]]s, viz it is a join of automorph-conjugate subgroups.

==Facts==

Any normal AC-generated subgroup is [[characteristic subgroup|characteristic]]. Thus, being AC-generated is a [[normal-to-characteristic subgroup property]].