Normality-preserving endomorphism-balanced 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-balanced subgroup of $G$ if, for every normality-preserving endomorphism $\sigma$ of $G$, the restriction of $\sigma$ to $H$ is a normality-preserving endomorphism of $H$.

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 $\to$ normality-preserving endomorphism	every normality-preserving endomorphism of $G$ restricts to a normality-preserving endomorphism of $H$	the balanced subgroup property for normality-preserving endomorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Prof of strictness (reverse implication failure)	Intermediate notions
Fully invariant direct factor
Fully invariant transitively normal subgroup

Weaker properties

Property	Meaning	Proof of implication	Prof of strictness (reverse implication failure)	Intermediate notions
Normality-preserving endomorphism-invariant subgroup	invariant under all normality-preserving endomorphisms			|FULL LIST, MORE INFO
Strictly characteristic subgroup	invariant under all surjective endomorphisms			|FULL LIST, MORE INFO
Characteristic subgroup	invariant under all automorphisms			|FULL LIST, MORE INFO
Normal subgroup	invariant under all inner automorphisms			|FULL LIST, MORE INFO