# 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]

## 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.

## Metaproperties

### Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT 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 $H \le K \le G$ are such that $H$ has finite double coset index in $K$ and $K$ has finite double coset index in $G$, it is not necessary that $H$ have finite double coset index in $G$. 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 condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H \le K \le G$ and $H$ has finite double coset index in $G$, then $H$ also has finite double coset index in $K$.

### 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 $H \le K \le G$ and $H$ has finite double coset index in $G$, then $K$ also has finite double coset index in $G$.

## 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