# Weakly normal-homomorph-containing subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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