# Conjugate-comparable subgroup

From Groupprops

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

## Contents

## Definition

A subgroup of a group is termed a **conjugate-comparable subgroup** if it is comparable with each of its conjugate subgroups, in other words, every conjugate subgroup to it either contains it or is contained in it.

## Relation with other properties

### Collapse to normality

- Any finite subgroup that is conjugate-comparable is normal.
- Any subgroup of finite index that is conjugate-comparable is normal.
- Any subgroup of a slender group, Artinian group, or periodic group that is conjugate-comparable is normal.

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Automorph-comparable subgroup | comparable to all its automorphic subgroups | |FULL LIST, MORE INFO | ||

Normal subgroup | equal to each conjuate subgroup | equal things can be compared | conjugate-comparable not implies normal | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Subgroup invariant under conjugation by a generating set |