Subgroup of finite double coset index

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.

Stronger properties

 * Weaker than::Subgroup of finite index
 * Weaker than::Subgroup of double coset index two

Weaker properties

 * Stronger than::Subgroup contained in finitely many intermediate subgroups:
 * Stronger than::Elliptic subgroup:

Metaproperties
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 $$G$$.

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

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

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.