Normality-preserving endomorphism-invariant subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Normality-preserving endomorphism-invariant subgroup, all facts related to Normality-preserving endomorphism-invariant subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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 arising from stronger properties or subgroup-defining functions

• All fully invariant subgroups, including the derived subgroup (commutator subgroup), as well as members of the derived series and lower central series, are normality-preserving endomorphism-invariant.
• The Fitting subgroup and solvable radical are both normality-preserving endomorphism-invariant. In fact, they are something stronger: weakly normal-homomorph-containing subgroups.

Examples in small finite groups

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

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

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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
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
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