# Weakly normal subgroup

From Groupprops

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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]

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 of a group is termed **weakly normal** if for any in , if implies .

(Here denotes the conjugate of by under the right action. We can use either the left or the right action for the definition).

## Relation with other properties

### Stronger properties

- Normal subgroup
- Pronormal subgroup:
`For full proof, refer: Pronormal implies weakly normal` - NE-subgroup:
*For proof of the implication, refer NE implies weakly normal and for proof of its strictness (i.e. the reverse implication being false) refer Weakly normal not implies NE*. - Paranormal subgroup:
*For proof of the implication, refer Paranormal implies weakly normal and for proof of its strictness (i.e. the reverse implication being false) refer Weakly normal not implies paranormal*. - Self-normalizing subgroup
- Weakly characteristic subgroup

### Weaker properties

- Intermediately subnormal-to-normal subgroup:
`For full proof, refer: Weakly normal implies intermediately subnormal-to-normal` - Subgroup with self-normalizing normalizer

## 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If are groups and is weakly normal in , then is weakly normal in . `For full proof, refer: Weak normality satisfies intermediate subgroup condition`