Commensurator of a subgroup

Definition
Suppose $$H$$ is a subgroup of a group $$G$$. The commensurator of $$H$$ in $$G$$ is the set of $$g \in G$$ such that $$H$$ and its conjugate $$gHg^{-1}$$ are defining ingredient::commensurable subgroups.

Facts

 * The commensurator of a subgroup is also a subgroup that contains the normalizer of the original subgroup.
 * The commensurator of a subgroup is the whole group if and only if the subgroup is a conjugate-commensurable subgroup.