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 commensurable subgroups.

Facts

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