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

Suppose \varphi:N \to G is a homomorphism of groups such that for any normal subgroup H of G contained in N, we have \varphi(H) is normal in G. Then, \varphi(N) \le N.

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 Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO
Characteristic subgroup invariant under all automorphisms (via strictly characteristic) (via strictly characteristic) Normality-preserving endomorphism-invariant subgroup, Strictly characteristic subgroup|FULL LIST, MORE INFO
Normal subgroup invariant under all inner automorphisms (via characteristic) (via characteristic) Normality-preserving endomorphism-invariant subgroup, Strictly characteristic subgroup|FULL LIST, MORE INFO