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

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

− | A [[subgroup property]] <math>p</math> is said to be '''upper join-closed''' if | + | 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>. |

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− | <math> | ||

==Relation with other metaproperties== | ==Relation with other metaproperties== | ||

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* [[Izable subgroup property]] | * [[Izable subgroup property]] | ||

+ | ===Weaker metaproperties=== | ||

+ | |||

+ | * [[Stronger than::Finite-upper join-closed subgroup property]] | ||

+ | * [[Stronger than::Permuting-upper join-closed subgroup property]] | ||

==Related notions== | ==Related notions== | ||

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===Normality=== | ===Normality=== | ||

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

===Central factor=== | ===Central factor=== | ||

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

## 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 metapropertyVIEW 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

## Contents

## Definition

### Definition with symbols

A subgroup property is said to be **upper join-closed** if given and are intermediate subgroups of containing (indexed by a nonempty set ) and satisfies in each , we have that satisfies in the join of subgroups .

## Relation with other metaproperties

### Stronger metaproperties

- Lower-intersection upper-join closed subgroup property
- LU-join closed subgroup property
- Upward-closed subgroup property
- Izable subgroup property

### Weaker metaproperties

## Related notions

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

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

## Properties satisfying it

### Normality

Normality is an upper join-closed subgroup property, viz, if and are intermediate subgroups such that and , then .

### Central factor

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