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 |