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

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.

Relation with other properties

Stronger properties

• Strongly closed subgroup

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 $H \le K \le G$ 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 $K$.