Maximal subgroup - Revision history

Vipul: /* Testing */

2008-11-21T22:48:43Z

Testing

← Older revision		Revision as of 22:48, 21 November 2008
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	==Testing==		==Testing==

	{{GAP command for ~~sgp~~\|		{{GAP command for subgroup property\|
	list = MaximalSubgroups\|		list = MaximalSubgroups\|
	classlist = ConjugacyClassesMaximalSubgroups\|		classlist = ConjugacyClassesMaximalSubgroups\|

2008-11-01T13:38:26Z

Relation with other properties

← Older revision		Revision as of 13:38, 1 November 2008
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	* [[Stronger than::Modular subgroup]]: {{proofat\|[[Maximal implies modular]]}}		* [[Stronger than::Modular subgroup]]: {{proofat\|[[Maximal implies modular]]}}
	* [[Stronger than::Pronormal subgroup]]: {{proofat\|[[Maximal implies pronormal]]}}		* [[Stronger than::Pronormal subgroup]]: {{proofat\|[[Maximal implies pronormal]]}}
			* [[Stronger than::Subgroup contained in finitely many intermediate subgroups]]

	===Property bifurcations===		===Property bifurcations===
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	[[:Category:Property bifurcations for maximal subgroups]]		[[:Category:Property bifurcations for maximal subgroups]]



	==Metaproperties==		==Metaproperties==

2008-10-30T23:10:15Z

← Older revision		Revision as of 23:10, 30 October 2008
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	{{GAP command for sgp\|		{{GAP command for sgp\|
	list = MaximalSubgroups\|		list = MaximalSubgroups\|
	classlist = ConjugacyClassesMaximalSubgroups}}		classlist = ConjugacyClassesMaximalSubgroups\|
			classreplist = MaximalSubgroupClassReps}}

	==Study of the notion==		==Study of the notion==

2008-10-30T22:55:18Z

← Older revision		Revision as of 22:55, 30 October 2008
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	The [[maximal operator]] is a [[subgroup property modifier]] that takes any subgroup property and gives out the property of being a subgroup that is maximal in the group with respect to that property.		The [[maximal operator]] is a [[subgroup property modifier]] that takes any subgroup property and gives out the property of being a subgroup that is maximal in the group with respect to that property.

			==Testing==

			{{GAP command for sgp\|
			list = MaximalSubgroups\|
			classlist = ConjugacyClassesMaximalSubgroups}}

	==Study of the notion==		==Study of the notion==

2008-10-02T13:43:37Z

Property theory

← Older revision		Revision as of 13:43, 2 October 2008
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	[[:Category:Property bifurcations for maximal subgroups]]		[[:Category:Property bifurcations for maximal subgroups]]

	~~==Property theory==~~

	~~===Closed or contraclosed===~~

	~~Given any [[expansive operator]] on the collection of subgroups of a group, a maximal subgroup is either closed with respect to that operator, or its closure is the whole group.~~

	* Every maximal subgroup is either [[normal subgroup\|normal]] or [[contranormal subgroup\|contranormal]].
	* Every maximal subgroup is either [[characteristic subgroup\|characteristic]] or [[contracharacteristic subgroup\|contracharacteristic]].
	* Every maximal subgroup is either [[normal subgroup\|normal]] or [[self-normalizing subgroup\|self-normalizing]].

	==Metaproperties==		==Metaproperties==

2008-10-02T13:42:41Z

Relation with other properties

← Older revision		Revision as of 13:42, 2 October 2008
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	===Weaker properties===		===Weaker properties===

	* [[NE-subgroup]]		* [[Stronger than::NE-subgroup]]
			* [[Stronger than::Modular subgroup]]: {{proofat\|[[Maximal implies modular]]}}
			* [[Stronger than::Pronormal subgroup]]: {{proofat\|[[Maximal implies pronormal]]}}

			===Property bifurcations===

			There are many pairs of properties such that every maximal subgroup of a group has ''exactly'' one of these properties. For a complete list, refer:

			[[:Category:Property bifurcations for maximal subgroups]]

	==Property theory==		==Property theory==

2008-09-02T20:26:36Z

Metaproperties

← Older revision		Revision as of 20:26, 2 September 2008
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	In general, it may not be true that the intersection of a maximal subgroup with another subgroup is maximal inside that subgroup. If a subgroup has the property that its intersection with every maximal subgroup (not containing it) is maximal in it, the subgroup is termed ''max-sensitive''.		In general, it may not be true that the intersection of a maximal subgroup with another subgroup is maximal inside that subgroup. If a subgroup has the property that its intersection with every maximal subgroup (not containing it) is maximal in it, the subgroup is termed ''max-sensitive''.

	~~{{upward-closed subgroup property}}~~

	==Property operators==		==Property operators==

2008-09-02T19:27:09Z

Relation with other properties

← Older revision		Revision as of 19:27, 2 September 2008
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	===Stronger properties===		===Stronger properties===

	* [[Subgroup of prime index]]		* [[Weaker than::Subgroup of prime index]]
	* [[Subgroup of index two]]		* [[Weaker than::Subgroup of index two]]
			* [[Weaker than::Subgroup of double coset index two]]

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

2008-05-07T23:50:20Z

15 revisions

← Older revision	Revision as of 23:50, 7 May 2008
(No difference)

2008-02-22T13:00:45Z

← Older revision		Revision as of 13:00, 22 February 2008
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	The property of being a maximal subgroup can be expressed in monadic second-order logic: there is no bigger subgroup between the given subgroup and the whole group.		The property of being a maximal subgroup can be expressed in monadic second-order logic: there is no bigger subgroup between the given subgroup and the whole group.

	{{obtainedbyapplyingthe\|maximal operator\|subgroup}}		{{obtainedbyapplyingthe\|maximal operator\|proper subgroup}}

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