Weakly normal-homomorph-containing subgroup

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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

Definition with symbols

A subgroup of a group is termed a weakly normal-homomorph-containing subgroup if is a normal subgroup of and the following holds:

Suppose is a homomorphism of groups such that for any normal subgroup of contained in , we have is normal in . Then, .

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal-homomorph-containing subgroup contains any homomorphic image that is normal in whole group normal-homomorph-containing implies weakly normal-homomorph-containing |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normality-preserving endomorphism-invariant subgroup invariant under all normality-preserving endomorphisms weakly normal-homomorph-containing implies normality-preserving endomorphism-invariant normality-preserving endomorphism-invariant not implies weakly normal-homomorph-containing |FULL LIST, MORE INFO
Strictly characteristic subgroup invariant under all surjective endomorphisms Weakly normal-homomorph-containing implies strictly characteristic Strictly characteristic not implies weakly normal-homomorph-containing |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