Weakly normal subgroup
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
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
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
- 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
- Intermediately subnormal-to-normal subgroup: For full proof, refer: Weakly normal implies intermediately subnormal-to-normal
- Subgroup with self-normalizing normalizer
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 condition
ABOUT 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