Conjugate-commensurable subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed conjugate-commensurable if it is commensurable with each of its conjugate subgroups. Equivalently, its commensurator in the whole group is the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed a conjugate-commensurable subgroup if, for any $g\in G$, $H\cap gHg^{-1}$ has finite index in both $H$ and $gHg^{-1}$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup				|FULL LIST, MORE INFO
subgroup of finite group
finite subgroup
subgroup of finite index				|FULL LIST, MORE INFO
nearly normal subgroup	subgroup of finite index in its normal closure
isomorph-commensurable subgroup	commensurable with every isomorphic subgroup
automorph-commensurable subgroup	commensurable with every automorphic subgroup