# Completely distinguished subgroup

From Groupprops

This article defines a subgroup property related to (or which arises in the context of): combinatorial group theory

View other subgroup properties related to combinatorial group theory|View other terms related to combinatorial group theory | View all subgroup properties

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed **completely distinguished** if for every surjective endomorphism from the group to itself, the subgroup equals its complete pre-image.

### Definition with symbols

A subgroup of a group is termed **completely distinguished** in if it satisfies the following equivalent conditions:

- For any surjective endomorphism ,
- is a distinguished subgroup (also termed a strictly characteristic subgroup) and is a Hopfian group

## Relation with other properties

### Weaker properties

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

surjective endomorphism-balanced subgroup | |FULL LIST, MORE INFO | |||

strictly characteristic subgroup | Template:Intermdiate notions shirt | |||

characteristic subgroup | Strictly characteristic subgroup, Surjective endomorphism-balanced subgroup|FULL LIST, MORE INFO |

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### Intersection-closedness

YES:This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closedABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness