# Weakly closed subgroup for a fusion system

From Groupprops

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

## Contents

## Definition

Suppose is a group of prime power order and is a fusion system on . Then a subgroup is termed **weakly closed** with respect to if for every morphism in , we have .

## Relation with other properties

### Stronger properties

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

- Weakly closed subgroup: Suppose are groups. is weakly closed in with respect to if, for any , implies that .
- Weakly closed subgroup of Sylow subgroup: The case where is a -Sylow subgroup of . This is related to fusion systems as follows: a subgroup of a -Sylow subgroup of a finite group is weakly closed in if and only if it is weakly closed for the fusion system induced by on .
- 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.