# Difference between revisions of "Subgroup contained in finitely many intermediate subgroups"

(New page: {{subgroup property}} ==Definition== ===Symbol-free definition=== A subgroup of a group is said to be '''contained in finitely many intermediate subgroups''' if the number of subgroups ...) |
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* If a normal subgroup is contained in only finitely many intermediate subgroups, it has finite index. This follows from the [[fourth isomorphism theorem]] and the fact that [[finitely many subgroups iff finite|a group having finitely many subgroups is finite]]. | * If a normal subgroup is contained in only finitely many intermediate subgroups, it has finite index. This follows from the [[fourth isomorphism theorem]] and the fact that [[finitely many subgroups iff finite|a group having finitely many subgroups is finite]]. | ||

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+ | ==Metaproperties== | ||

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+ | {{intransitive}} | ||

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+ | If <math>H \le K \le G</math> are groups such that <math>K</math> is contained in only finitely many intermediate subgroups of <math>G</math> and <math>H</math> is contained in only finitely many intermediate subgroups of <math>K</math>, it may well happen that <math>H</math> is contained in infinitely many intermediate subgroups of <math>G</math>. | ||

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+ | {{intsubcondn}} | ||

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+ | If <math>H \le K \le G</math> are groups such that <math>H</math> is contained in only finitely many intermediate subgroups of <math>G</math>, then <math>H</math> is also contained in only finitely many intermediate subgroups of <math>K</math>. | ||

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+ | {{upward-closed}} | ||

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+ | If <math>H \le K \le G</math> are groups such that <math>H</math> is contained in only finitely many intermediate subgroups of <math>G</math>, then <math>K</math> is also contained in only finitely many intermediate subgroups of <math>G</math>. |

## Latest revision as of 13:42, 1 November 2008

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is said to be **contained in finitely many intermediate subgroups** if the number of subgroups of the whole group containing that subgroup is finite.

### Definition with symbols

A subgroup of a group is said to be **contained in finitely many intermediate subgroups** if the number of subgroups of such that is finite.

## Relation with other properties

### Stronger properties

## Facts

- If a normal subgroup is contained in only finitely many intermediate subgroups, it has finite index. This follows from the fourth isomorphism theorem and the fact that a group having finitely many subgroups is finite.

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

If are groups such that is contained in only finitely many intermediate subgroups of and is contained in only finitely many intermediate subgroups of , it may well happen that is contained in infinitely many intermediate subgroups of .

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If are groups such that is contained in only finitely many intermediate subgroups of , then is also contained in only finitely many intermediate subgroups of .

### Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group

View other upward-closed subgroup properties

If are groups such that is contained in only finitely many intermediate subgroups of , then is also contained in only finitely many intermediate subgroups of .