# Difference between revisions of "Normal subgroup having no nontrivial homomorphism to its quotient group"

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]

## Definition

A subgroup $N$ of a group $G$ is a normal subgroup having no nontrivial homomorphism to its quotient group if $N$ is a normal subgroup of $G$ and there is no nontrivial homomorphism from $N$ to its quotient group $G/N$.

## Metaproperties

### Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the 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 subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

For full proof, refer: No nontrivial homomorphism to quotient group is not 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

### Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H \le K \le G$, with $H$ normal in $G$ and no nontrivial homomorphism from $H$ to $G/H$, $H$ is also normal in $K$ with no nontrivial homomorphism from $H$ to $K/H$.

For full proof, refer: No nontrivial homomorphism to quotient group satisfies intermediate subgroup condition