Weakly closed subgroup
This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
View other such properties
Suppose . Then, is termed weakly closed in relative to if, for any such that , we have .
There is a related notion of weakly closed subgroup for a fusion system.
Relation with other properties
- Normalizer-relatively normal subgroup: For full proof, refer: Weakly closed implies normalizer-relatively normal
- Relatively normal subgroup: For full proof, refer: Weakly closed implies normal in middle subgroup
- Conjugation-invariantly relatively normal subgroup when the big group is a finite group: For full proof, refer: Weakly closed implies conjugation-invariantly relatively normal in finite group
- Weakly closed implies normal in middle subgroup: If and is weakly closed in relative to , then is a normal subgroup of .
- Weakly normal implies weakly closed in intermediate nilpotent: If , with a weakly normal subgroup of , and a nilpotent group, then is a weakly closed subgroup of .