Normality-preserving endomorphism-invariant subgroup

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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]

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Definition

A subgroup $H$ of a group $G$ is termed a normality-preserving endomorphism-invariant subgroup if, for every normality-preserving endomorphism $\sigma$ of $G$, $\sigma (H)$ is contained in $H$.

Formalisms

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

Function restriction expression	$H$ is a normality-preserving endomorphism-invariant subgroup of $G$ if ...	This means that the property is ...	Comments
normality-preserving endomorphism $\to$ function	every normality-preserving endomorphism of $G$ sends every element of $H$ to within $H$	the invariance property for normality-preserving endomorphisms
normality-preserving endomorphism $\to$ endomorphism	every normality-preserving endomorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for normality-preserving endomorphisms; i.e., it is the invariance property for normality-preserving endomorphism, which is a property stronger than the property of being an endomorphism

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Fully invariant subgroup	invariant under all endomorphisms	fully invariant implies normality-preserving endomorphism-invariant	normality-preserving endomorphism-invariant not implies fully invariant	|FULL LIST, MORE INFO
Normal-homomorph-containing subgroup	any homomorphic image of the subgroup that's normal in the whole group is contained in the subgroup	normal-homomorph-containing implies normality-preserving endomorphism-invariant	normality-preserving endomorphism-invariant not implies normal-homomorph-containing	|FULL LIST, MORE INFO
Normal-subhomomorph-containing subgroup	any homomorphic image of a subgroup that's normal in the whole group is contained in the subgroup	(via normal-homomorph-containing)	(via normal-homomorph-containing)	|FULL LIST, MORE INFO
Weakly normal-homomorph-containing subgroup	image of subgroup under a homomorphism that sends normal subgroups inside it to normal subgroups is normal	weakly normal-homomorph-containing implies normality-preserving endomorphism-invariant	normality-preserving endomorphism-invariant not implies weakly normal-homomorph-containing	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Strictly characteristic subgroup	invariant under all surjective endomorphisms	normality-preserving endomorphism-invariant implies strictly characteristic	strictly characteristic not implies normality-preserving endomorphism-invariant	|FULL LIST, MORE INFO
Characteristic subgroup	invariant under all automorphisms	(via strictly characteristic)	(via strictly characteristic)	|FULL LIST, MORE INFO
Normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	|FULL LIST, MORE INFO