Weakly normal-homomorph-containing subgroup

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

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Weakly normal-homomorph-containing subgroup, all facts related to Weakly normal-homomorph-containing subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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

Relation with other properties

Stronger properties

• Normal-homomorph-containing subgroup

Weaker properties

• Strictly characteristic subgroup: For full proof, refer: Weakly normal-homomorph-containing implies strictly characteristic
• Characteristic subgroup