Normal subgroup having no nontrivial homomorphism to its quotient group

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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.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness Intermediate notions
normal Sylow subgroup the whole group is a finite group and the subgroup is both normal and a Sylow subgroup. |FULL LIST, MORE INFO
normal Hall subgroup the whole group is a finite group and the subgroup is both normal and a Hall subgroup -- its order and index are relatively prime. |FULL LIST, MORE INFO
[Weaker than::fully invariant direct factor]] both a fully invariant subgroup and a direct factor of the whole group. |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness Intermediate notions
homomorph-containing subgroup contains every homomorphic image of itself in the whole group. |FULL LIST, MORE INFO
quotient-subisomorph-containing subgroup |FULL LIST, MORE INFO
fully invariant subgroup invariant under all endomorphisms (via homomorph-containing) (via homomorph-containing) Homomorph-containing subgroup, Intermediately fully invariant subgroup, Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms (via fully invariant) (via fully invariant) Fully invariant subgroup, Homomorph-containing subgroup, Intermediately fully invariant subgroup, Intermediately strictly characteristic subgroup, Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
normal subgroup (by definition) (via characteristic) Fully invariant subgroup, Homomorph-containing subgroup|FULL LIST, MORE INFO

Related properties

Other incomparable properties

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