# Normality-preserving endomorphism-invariant subgroup

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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 $H$. A normality-preserving endomorphism is an endomorphism with the property that the image of any normal subgroup is normal.

## Examples

### Extreme examples

• The trivial subgroup is normality-preserving endomorphism-invariant in any group.
• Every group is normality-preserving endomorphism-invariant in itself.

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property normality-preserving endomorphism-invariant subgroup.

Group partSubgroup partQuotient part
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2

Below are some examples of a proper nontrivial subgroup that does not satisfy the property normality-preserving endomorphism-invariant subgroup.

Group partSubgroup partQuotient part
Diagonally embedded Z4 in direct product of Z8 and Z2Direct product of Z8 and Z2Cyclic group:Z4Cyclic group:Z4
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]

## 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 Weakly normal-homomorph-containing subgroup|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
Normality-preserving endomorphism-balanced subgroup any normality-preserving endomorphism of the whole group restricts to a normality-preserving endomorphism of the subgroup |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
direct projection-invariant subgroup invariant under all projections to direct factors normality-preserving endomorphism-invariant implies direct projection-invariant direct projection-invariant not implies normality-preserving endomorphism-invariant |FULL LIST, MORE INFO
finite direct power-closed characteristic subgroup in any finite direct power of the whole group, the corresponding power of the subgroup is characteristic normality-preserving endomorphism-invariant implies finite direct power-closed characteristic finite direct power-closed characteristic not implies normality-preserving endomorphism-invariant |FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms (via strictly characteristic) (via strictly characteristic) Finite direct power-closed characteristic subgroup|FULL LIST, MORE INFO
normal subgroup invariant under all inner automorphisms (via characteristic) (via characteristic) Finite direct power-closed characteristic subgroup|FULL LIST, MORE INFO