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]

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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 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

Effect of property operators

The intermediately operator

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