# Completely distinguished subgroup

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

## 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 $H$ of a group $G$ is termed completely distinguished in $G$ if it satisfies the following equivalent conditions:

• For any surjective endomorphism $f:G \to G$, $f^{-1}(H) = H$
• $H$ is a distinguished subgroup (also termed a strictly characteristic subgroup) and $G/H$ is a Hopfian group

## Relation with other properties

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
strictly characteristic subgroup Template:Intermdiate notions shirt

## 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 transitive
ABOUT 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-closed
ABOUT 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