Modular subgroup - Revision history

Vipul: /* Definition */

2017-04-16T16:07:19Z

Definition

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	==Definition==		==Definition==

	~~===Symbol-free definition===~~		A [[subgroup]] of a [[group]] is termed a '''modular subgroup''' if it is a modular element in the [[lattice of subgroups]]. Explicitly, a [[subgroup]] <math>A</math> of a group <math>G</math> is termed a '''modular subgroup''' if for any subgroups <math>B</math> and <math>C</math> of <math>G</math> such that <math>A \le C</math>:

	~~A [[subgroup]] of a [[group]] is termed a '''modular subgroup''' if it is a modular element in the [[lattice of subgroups]].~~		<math> \langle A, B \cap C \rangle = \langle A,B \rangle \cap C</math>

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

	~~A [[subgroup]] <math>A</math> of a group <math>G</math> is termed a '''modular subgroup''' if for any subgroups <math>B</math> and <math>C</math> of <math>G</math> such that <math>A \le C</math>:~~

	<math> \langle A, B \cap C \rangle = \langle A,B \rangle \cap C</math>

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

Vipul at 01:33, 26 November 2012

2012-11-26T01:33:31Z

← Older revision		Revision as of 01:33, 26 November 2012
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			{{subgroup property}}
	{{lattice-determined subgroup property}}		{{lattice-determined subgroup property}}

	{{variationof\|normality}}		{{variationof\|normality}}

Vipul: /* Relation with other properties */

2012-11-26T01:33:01Z

Relation with other properties

← Older revision		Revision as of 01:33, 26 November 2012
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::~~Normal~~ subgroup]]		{\| class="sortable" border="1"
	* [[~~Weaker than::Permutable subgroup~~]]~~: {{proofofstrictimplicationat~~\|[[~~Permutable~~ implies modular]]\|[[~~Modular~~ not implies permutable]]}}		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::~~Distributive~~ subgroup]]		\|-
			\| [[Weaker than::permutable subgroup]] \|\| permutes (commutes) with every subgroup, i.e., its [[product of subgroups\|product]] with any subgroup is a subgroup \|\| [[permutable implies modular]] (proof relies on the [[modular property of groups]]) \|\| [[modular not implies permutable]] \|\| {{intermediate notions short\|modular subgroup\|permutable subgroup}}
	~~The proof for permutable subgroups~~ (~~and hence~~, ~~for~~ normal ~~subgroups~~) ~~follows from the~~ [[modular ~~property of groups~~]].		\|-
			\| [[Weaker than::normal subgroup]] \|\| permutes (commutes) with every element, i.e., its left cosets are the same as its right cosets \|\| ([[normal implies permutable\|via permutable]]) \|\| (via permutable) \|\| {{intermediate notions short\|modular subgroup\|normal subgroup}}
			\|-
			\| [[Weaker than::maximal subgroup]] \|\| proper subgroup not contained in any bigger proper subgroup \|\| [[maximal implies modular]] \|\| obvious from the fact that there are normal subgroups that are not maximal, and normal implies modular \|\| {{intermediate notions short\|modular subgroup\|maximal subgroup}}
			\|}
	===Weaker properties===		===Weaker properties===

	* [[Stronger than::~~Nilpotent~~ quotient-by-core subgroup]]		{\| class="sortable" border="1"
	* [[Stronger than::~~Permodular~~ subgroup]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::nilpotent quotient-by-core subgroup]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::permodular subgroup]] \|\| \|\| \|\| \|\|
			\|}

	==Metaproperties==		==Metaproperties==

Vipul: /* Metaproperties */

2008-12-15T19:46:47Z

Metaproperties

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	{{proofat\|[[Modularity satisfies intermediate subgroup condition]]}}		{{proofat\|[[Modularity satisfies intermediate subgroup condition]]}}

	~~{{join-closed}}~~

	~~A join of modular subgroups is modular. This follows from a general lattice-theoretic fact that a join of modular elements is modular.~~

	~~{{proofat\|[[Modularity is strongly join-closed]]}}~~

Vipul: /* Metaproperties */

2008-12-15T19:30:28Z

Metaproperties

← Older revision		Revision as of 19:30, 15 December 2008
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	Thus, <math>H</math> is also modular in <math>K</math>.		Thus, <math>H</math> is also modular in <math>K</math>.

	~~In fact, it's a general lattice-theoretic fact that any modular element is also modular in any interval sublattice.~~		{{proofat\|[[Modularity satisfies intermediate subgroup condition]]}}

	{{join-closed}}		{{join-closed}}

	A join of modular subgroups is modular. This follows from a general lattice-theoretic fact that a join of modular elements is modular.		A join of modular subgroups is modular. This follows from a general lattice-theoretic fact that a join of modular elements is modular.

			{{proofat\|[[Modularity is strongly join-closed]]}}

Vipul at 13:09, 2 October 2008

2008-10-02T13:09:11Z

← Older revision		Revision as of 13:09, 2 October 2008
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	A [[subgroup]] <math>A</math> of a group <math>G</math> is termed a '''modular subgroup''' if for any subgroups <math>B</math> and <math>C</math> of <math>G</math> such that <math>A \le C</math>:		A [[subgroup]] <math>A</math> of a group <math>G</math> is termed a '''modular subgroup''' if for any subgroups <math>B</math> and <math>C</math> of <math>G</math> such that <math>A \le C</math>:

	<math> <A, B \cap C > = <A,B> \cap C</math>		<math> \langle A, B \cap C \rangle = \langle A,B \rangle \cap C</math>

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

	* [[Normal subgroup]]		* [[Weaker than::Normal subgroup]]
	* [[Permutable subgroup]]: {{proofofstrictimplicationat\|[[Permutable implies modular]]\|[[Modular not implies permutable]]}}		* [[Weaker than::Permutable subgroup]]: {{proofofstrictimplicationat\|[[Permutable implies modular]]\|[[Modular not implies permutable]]}}
	* [[Distributive subgroup]]		* [[Weaker than::Distributive subgroup]]

	The proof for permutable subgroups (and hence, for normal subgroups) follows from the [[modular property of groups]].		The proof for permutable subgroups (and hence, for normal subgroups) follows from the [[modular property of groups]].
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	===Weaker properties===		===Weaker properties===

	* [[Nilpotent quotient-by-core subgroup]]		* [[Stronger than::Nilpotent quotient-by-core subgroup]]
	* [[Permodular subgroup]]		* [[Stronger than::Permodular subgroup]]

	==Metaproperties==		==Metaproperties==

Vipul: 5 revisions

2008-05-07T23:51:21Z

5 revisions

← Older revision	Revision as of 23:51, 7 May 2008
(No difference)

Vipul: /* Weaker properties */

2007-09-04T08:54:29Z

Weaker properties

← Older revision		Revision as of 08:54, 4 September 2007
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	* [[Nilpotent quotient-by-core subgroup]]		* [[Nilpotent quotient-by-core subgroup]]
			* [[Permodular subgroup]]

	==Metaproperties==		==Metaproperties==

Vipul: /* Relation with other properties */

2007-09-04T08:54:00Z

Relation with other properties

← Older revision		Revision as of 08:54, 4 September 2007
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	* [[Normal subgroup]]		* [[Normal subgroup]]
	* [[Permutable subgroup]]		* [[Permutable subgroup]]: {{proofofstrictimplicationat\|[[Permutable implies modular]]\|[[Modular not implies permutable]]}}
	* [[Distributive subgroup]]		* [[Distributive subgroup]]

Vipul: /* Weaker properties */

2007-04-03T10:30:11Z

Weaker properties

← Older revision		Revision as of 10:30, 3 April 2007
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	===Weaker properties===		===Weaker properties===

	* [[[Nilpotent quotient-by-core subgroup]]		* [[Nilpotent quotient-by-core subgroup]]

	==Metaproperties==		==Metaproperties==