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 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.

	Group part	Subgroup part	Quotient part
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2

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

	Group part	Subgroup part	Quotient part
Diagonally embedded Z4 in direct product of Z8 and Z2	Direct product of Z8 and Z2	Cyclic group:Z4	Cyclic group:Z4

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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	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