# Subgroup of finite double coset index

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

A **subgroup of finite double coset index** is a subgroup of a group whose double coset space is finite -- in other words, the subgroup has only finitely many double cosets in the group.

## Relation with other properties

### Stronger properties

### Weaker properties

- Subgroup contained in finitely many intermediate subgroups:
*For proof of the implication, refer Finite double coset index implies finitely many intermediate subgroups and for proof of its strictness (i.e. the reverse implication being false) refer Finitely many intermediate subgroups not implies finite double coset index*. - Elliptic subgroup:
*For proof of the implication, refer Finite double coset index implies elliptic and for proof of its strictness (i.e. the reverse implication being false) refer Elliptic not implies finite double coset index*.

## 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 such that has finite double coset index in and has finite double coset index in , it is *not* necessary that have finite double coset index in . `For full proof, refer: Finite double coset index is not transitive`

### 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 and has finite double coset index in , then also has finite double coset index in .

### 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 and has finite double coset index in , then also has finite double coset index in .

## Effect of property operators

### Left transiter

It is true that any subgroup of finite index inside a subgroup of finite double coset index again has finite double coset index. `For full proof, refer: Finite index in finite double coset index implies finite double coset index`