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]

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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

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	No	no nontrivial homomorphism to quotient group is not transitive	It is possible to have groups $H\leq K\leq G$ such that $H$ is normal in $K$ with no nontrivial homomorphism from $H$ to $K/H$, and $K$ is normal in $G$ with no nontrivial homomorphism from $K$ to $G/K$, but there does exist a nontrivial homomorphism from $H$ to $G/H$.
trim subgroup property	Yes	(obvious reasons)	In any group, the trivial subgroup and the whole group satisfy the condition of being normal subgroups with no nontrivial homomorphism to their respective quotient groups.
intermediate subgroup condition	Yes	no nontrivial homomorphism to quotient group satisfies intermediate subgroup condition	If $H\leq K\leq G$ are groups such that $H$ is normal in $G$ and there is no nontrivial homomorphism from $H$ to $G/H$, $H$ is also normal in $K$ with no nontrivial homomorphism from $H$ to $K/H$.

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
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.		homomorph-containing not implies no nontrivial homomorphism to quotient 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)	|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms	(via fully invariant)	(via fully invariant)	Fully invariant subgroup|FULL LIST, MORE INFO
normal subgroup		(by definition)	(via characteristic)	Fully invariant subgroup|FULL LIST, MORE INFO

Related properties

• Normal subgroup having no nontrivial homomorphism from its quotient group
• Normal subgroup having no common composition factor with its quotient group

Other incomparable properties

• Complemented normal subgroup: For full proof, refer: No nontrivial homomorphism to quotient group not implies complemented normal

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 $H\leq K\leq G$, 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