# Commensurator

In group theory, a branch of abstract algebra, the commensurator of a subgroup H of a group G is a specific subgroup of G.

## Definition

The commensurator of a subgroup H of a group G, denoted commG(H) or by some comm(H),[1] is the set of all elements g of G that conjugate H and leave the result commensurable with H. In other words,

${\displaystyle \mathrm {comm} _{G}(H)=\{g\in G:gHg^{-1}\cap H{\text{ has finite index in both }}H{\text{ and }}gHg^{-1}\}.}$[2]

## Properties

• commG(H) is a subgroup of G.
• commG(H) = G for any compact open subgroup H.