# Automorph-commensurable subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup

## 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 of a group is termed an **automorph-commensurable subgroup** if, for any automorphism of , has finite index in both and .

## 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 |