# Weakly normal subgroup

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[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]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

## Definition

### Symbol-free definition

A subgroup of a group is termed weakly normal if whenever any conjugate of it is contained in its normalizer, the conjugate is actually contained in the subgroup itself. Equivalently, a subgroup is weakly normal if it is a weakly closed subgroup inside its normalizer relative to the whole group.

### Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly normal if for any $g$ in $G$, if $H^g \le N_G(H)$ implies $H^g \le H$.

(Here $H^g = g^{-1}Hg$ denotes the conjugate of $H$ by $g$ under the right action. We can use either the left or the right action for the definition).

## Metaproperties

### Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H \le K \le G$ are groups and $H$ is weakly normal in $G$, then $H$ is weakly normal in $K$. For full proof, refer: Weak normality satisfies intermediate subgroup condition