# Conjugate-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 normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup

## 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 of a group is termed a **conjugate-commensurable subgroup** if, for any , 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 |
---|---|---|---|---|

normal subgroup | Nearly normal subgroup|FULL LIST, MORE INFO | |||

subgroup of finite group | ||||

finite subgroup | ||||

subgroup of finite index | Automorph-commensurable subgroup, Nearly normal subgroup|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 |