# Normality-preserving endomorphism-balanced subgroup

From Groupprops

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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 of a group is termed a **normality-preserving endomorphism-balanced subgroup** of if, for every normality-preserving endomorphism of , the restriction of to is a normality-preserving endomorphism of .

## 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 | is a normality-preserving endomorphism-invariant subgroup of if ... | This means that the property is ... | Comments |
---|---|---|---|

normality-preserving endomorphism normality-preserving endomorphism | every normality-preserving endomorphism of restricts to a normality-preserving endomorphism of | 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 |