# Weakly abnormal subgroup

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

### Symbol-free definition

A subgroup of a group is termed **weakly abnormal** or **upward-closed self-normalizing** or **intermediately contranormal** if it satisfies the following equivalent conditions:

- Every element of the group lies in the closure of this subgroup under the action by conjugation by the cyclic subgroup generated by that element.
- Every subgroup containing that subgroup is a self-normalizing subgroup of the whole group.
- The subgroup is a contranormal subgroup in every intermediate subgroup.

### Definition with symbols

A subgroup of a group is termed **weakly abnormal** or **upward-closed self-normalizing** or **intermediately contranormal** if it satisfies the following equivalent conditions:

- Given any , . Here is the smallest subgroup of containing , which is closed under the action by conjugation by the cyclic subgroup generated by
- If , then is a self-normalizing subgroup of .
- If , then is a contranormal subgroup of .

### Equivalence of definitions

`For full proof, refer: Equivalence of definitions of weakly abnormal subgroup`

## Formalisms

### In terms of the upward-closure operator

This property is obtained by applying the upward-closure operator to the property: self-normalizing subgroup

View other properties obtained by applying the upward-closure operator

is weakly abnormal in if and only if every subgroup of containing is self-normalizing.

### In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: contranormal subgroup

View other properties obtained by applying the intermediately operator

is weakly abnormal in if and only if is contranormal in every intermediate subgroup.

## Relation with other properties

### Stronger properties

- Non-normal maximal subgroup
- Abnormal subgroup

### Weaker properties

- Weakly pronormal subgroup
- Self-normalizing subgroup
- Contranormal subgroup
- Paracharacteristic subgroup
- Paranormal subgroup
- Polycharacteristic subgroup
- Polynormal subgroup

## 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 a subgroup is weakly abnormal in the whole group, it is also weakly abnormal in every intermediate subgroup. `For full proof, refer: Weak abnormality satisfies intermediate subgroup condition]]`

### Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group

View other upward-closed subgroup properties

If a subgroup is weakly abnormal in the whole group, then every subgroup containing it is also weakly abnormal in the whole group. `For full proof, refer: Weak abnormality is upward-closed`