# Subgroup with self-normalizing normalizer

From Groupprops

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]

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **subgroup with self-normalizing normalizer** if its normalizer in the whole group is a self-normalizing subgroup of the whole group.

## Relation with other properties

### Stronger properties

- Normal subgroup
- Self-normalizing subgroup
- Intermediately subnormal-to-normal subgroup:
`For full proof, refer: Normalizer of intermediately subnormal-to-normal implies self-normalizing` - Procharacteristic subgroup
- Pronormal subgroup
- Subgroup with abnormal normalizer
- Weakly procharacteristic subgroup
- Weakly pronormal subgroup
- Subgroup with weakly abnormal normalizer
- Paracharacteristic subgroup
- Paranormal subgroup
- Polycharacteristic subgroup
- Polynormal subgroup

### Opposite properties

A subgroup can have a self-normalizing normalizer *and* satisfy one of the properties below *only if* it is normal.

- 2-hypernormalized subgroup: A subgroup whose normalizer is a normal subgroup.
- Finitarily hypernormalized subgroup: A subgroup such that repeatedly taking normalizers reaches the whole group in finitely many steps.
- Hypernormalized subgroup: A subgroup such that repeatedly taking normalizers reaches the whole group eventually (possibly, using a transfinite number of steps).

### Incomparable properties

- Subnormal subgroup
- 2-subnormal subgroup:
`Further information: Abnormal normalizer and 2-subnormal not implies normal`

## Metaproperties

### Intermediate subgroup condition

NO:This subgroup property doesnotsatisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some 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 SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition

## Effect of property operators

### The intermediately operator

*Applying the intermediately operator to this property gives*: intermediately subnormal-to-normal subgroup