# Weakly closed subgroup

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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.
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## 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.

## 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 $K$.