Subgroup with self-normalizing normalizer
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]
Relation with other 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
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).
- Subnormal subgroup
- 2-subnormal subgroup: Further information: Abnormal normalizer and 2-subnormal not implies normal
Intermediate subgroup condition
NO: This subgroup property does not satisfy 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 condition
ABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition