Subgroup of finite double coset index

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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

• Subgroup of finite index
• Subgroup of double coset index two

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 is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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If $H\leq K\leq G$ 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.
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If $H\leq K\leq G$ 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
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If $H\leq K\leq G$ 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