# Weakly closed subgroup for a fusion system

This article defines a property that can be evaluated for a group of prime power order, equipped with a fusion system
View other such properties

## Definition

Suppose $P$ is a group of prime power order and $\mathcal{F}$ is a fusion system on $P$. Then a subgroup $Q \le P$ is termed weakly closed with respect to $\mathcal{F}$ if for every morphism $\varphi: Q \to P$ in $\mathcal{F}$, we have $\varphi(Q) = Q$.

## Relation with other properties

### Related subgroup properties and subgroup-of-subgroup properties

• Weakly closed subgroup: Suppose $H \le K \le G$ are groups. $H$ is weakly closed in $K$ with respect to $G$ if, for any $g \in G$, $gHg^{-1} \le K$ implies that $gHg^{-1} \le H$.
• Weakly closed subgroup of Sylow subgroup: The case where $K$ is a $p$-Sylow subgroup of $G$. This is related to fusion systems as follows: a subgroup of a $p$-Sylow subgroup $K$ of a finite group $G$ is weakly closed in $K$ if and only if it is weakly closed for the fusion system induced by $G$ on $K$.
• Sylow-relatively weakly closed subgroup: A subgroup of a group of prime power order that is a weakly closed subgroup in any group containing the bigger group as a Sylow subgroup.
• Fusion system-relatively weakly closed subgroup: A subgroup of a group of prime power order that is weakly closed in the fusion system sense).for any fusion system on the bigger group.