# Surjective endomorphism-balanced subgroup

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]
If the ambient group is a finite group, this property is equivalent to the property: characteristic subgroup
View other properties finitarily equivalent to characteristic subgroup | View other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

## Definition

### Symbol-free definition

A subgroup of a group is termed surjective endomorphism-balanced if every surjective endomorphism of the whole group restricts to a surjective endomorphism of the subgroup.

### Definition with symbols

A subgroup $H$ of a group $G$ is termed surjective endomorphism-balanced if for any surjective endomorphism $\sigma$ of $G$, $\sigma(H) = H$.

## Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

In the function restriction formalism, the property of being highly strictly characteristic is expressed as:

Surjective endomorphism $\to$ surjective endomorphism

## 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

Transitivity follows directly from the fact that the property is a balanced subgroup property.

Further information: Balanced implies transitive

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties