Automorph-commensurable subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed an automorph-commensurable subgroup if it is commensurable with all its automorphic subgroups.

Definition with symbols

A subgroup $H$ of a group $G$ is termed an automorph-commensurable subgroup if, for any automorphism $\sigma$ of $G$, $H \cap \sigma(H)$ has finite index in both $H$ and $\sigma(H)$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Characteristic subgroup
Finite subgroup
Subgroup of finite group
Subgroup of finite index
Isomorph-commensurable subgroup

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Conjugate-commensurable subgroup