Normality-preserving endomorphism-invariant subgroup: Difference between revisions

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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 $σ$ of $G$ , $σ (H)$ 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]
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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

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 {{wikilocal}}
+==Definition==
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''normality-preserving endomorphism-invariant subgroup''' if, for every [[normality-preserving endomorphism]] <math>\sigma</math> of <math>G</math>, <math>\sigma(H)</math> is contained in <math>H</math>. A normality-preserving endomorphism is an endomorphism with the property that the image of any [[normal subgroup]] is normal.
+==Examples==
-{{subgroup property}}
+===Extreme examples===
+* The trivial subgroup is normality-preserving endomorphism-invariant in any group.
+* Every group is normality-preserving endomorphism-invariant in itself.
-==Definition==
+===Examples arising from stronger properties or subgroup-defining functions===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''normality-preserving endomorphism-invariant subgroup''' if, for every [[normality-preserving endomorphism]] <math>\sigma</math> of <math>G</math>, <math>\sigma(H)</math> is contained in <math>H</math>.
+* All [[fully invariant subgroup]]s, 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 subgroup]]s.
-==Formalisms==
+===Examples in small finite groups===
-{{frexp}}
+{{subgroup property see examples embed|normality-preserving endomorphism-invariant subgroup}}
-{| class="sortable" border="1"
+{{subgroup property}}
-! Function restriction expression !! <math>H</math> is a normality-preserving endomorphism-invariant subgroup of <math>G</math> if ... !! This means that the property is ...
-|-
-| {{invariance-short|normality-preserving endomorphism}}
-|-
-| {{endo-invariance-short|normality-preserving endomorphism}}
-|}
 ==Relation with other properties==