Weakly normal-homomorph-containing subgroup

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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, $\varphi (N)\leq N$.

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	|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