Difference between revisions of "Upper join-closed subgroup property"

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(Normality)
 
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===Definition with symbols===
 
===Definition with symbols===
  
A [[subgroup property]] <math>p</math> is said to be '''upper join-closed''' if whenever <math>H \le G</math> and <math>K_1,K_2</math> are intermediate subgroups of <math>G</math> containing <math>H</math>, then:
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A [[subgroup property]] <math>p</math> is said to be '''upper join-closed''' if given <math>H \le G</math> and <math>K_i, i \in I</math> are intermediate subgroups of <math>G</math> containing <math>H</math> (indexed by a nonempty set <math>I</math>) and <math>H</math> satisfies <math>p</math> in each <math>K_i</math>, we have that <math>H</math> satisfies <math>p</math> in the [[defining ingredient::join of subgroups]] <math>\langle K_i \rangle_{i \in I}</math>.
 
 
<math>H</math> satisfies <math>p</math> in <math>K_1</math> and <math>H</math> satisfies <math>p</math> in <math>K_2 \implies H</math> satisfies <math>p</math> in <math><K_1,K_2></math>.
 
  
 
==Relation with other metaproperties==
 
==Relation with other metaproperties==
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* [[Izable subgroup property]]
 
* [[Izable subgroup property]]
  
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===Weaker metaproperties===
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* [[Stronger than::Finite-upper join-closed subgroup property]]
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* [[Stronger than::Permuting-upper join-closed subgroup property]]
 
==Related notions==
 
==Related notions==
  

Latest revision as of 21:11, 12 May 2010

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions
This article is about a general term. A list of important particular cases (instances) is available at Category:Upper join-closed subgroup properties

Definition

Definition with symbols

A subgroup property p is said to be upper join-closed if given H \le G and K_i, i \in I are intermediate subgroups of G containing H (indexed by a nonempty set I) and H satisfies p in each K_i, we have that H satisfies p in the join of subgroups \langle K_i \rangle_{i \in I}.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties

Related notions

Given a subgroup property p that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup H of G associate a unique largest subgroup M containing H for which H satisfies p in M.

Such a subgroup property is termed an izable subgroup property and the M that we get is termed the izing subgroup of H for that subgroup property.

Properties satisfying it

Normality

Normality is an upper join-closed subgroup property, viz, if H \le G and K_1, K_2 are intermediate subgroups such that H \triangleleft K_1 and H \triangleleft K_2, then H \triangleleft <K_1,K_2>.

Central factor

The property of being a central factor is also upper join-closed, in fact, it is izable.