Weakly closed subgroup

Definition
Suppose $$H \le K \le G$$. Then, $$H$$ is termed weakly closed in $$K$$ relative to $$G$$ if, for any $$g \in G$$ such that $$gHg^{-1} \le K$$, we have $$gHg^{-1} \le H$$.

There is a related notion of weakly closed subgroup for a fusion system.

Stronger properties

 * Weaker than::Strongly closed subgroup

Weaker properties

 * Stronger than::Normalizer-relatively normal subgroup:
 * Stronger than::Relatively normal subgroup:
 * Stronger than::Conjugation-invariantly relatively normal subgroup when the big group is a finite group:

Facts

 * Weakly closed implies normal in middle subgroup: If $$H \le K \le G$$ and $$H$$ is weakly closed in $$K$$ relative to $$G$$, then $$H$$ is a normal subgroup of $$K$$.
 * Weakly normal implies weakly closed in intermediate nilpotent: If $$H \le K \le G$$, with $$H$$ a weakly normal subgroup of $$G$$, and $$K$$ a nilpotent group, then $$H$$ is a weakly closed subgroup of $$K$$.