Weakly closed subgroup
From Groupprops
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
Contents
Definition
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
Stronger properties
Weaker 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
Facts
- 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
.