# Normality-preserving endomorphism-balanced subgroup

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## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 $H$ 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 Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO
Characteristic subgroup invariant under all automorphisms Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO
Normal subgroup invariant under all inner automorphisms Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO