# 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

## Contents

## Definition

Suppose . Then, is termed **weakly closed** in relative to if, for any such that , we have .

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

## Relation with other properties

### Stronger properties

### 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 and is weakly closed in relative to , then is a normal subgroup of .
- Weakly normal implies weakly closed in intermediate nilpotent: If , with a weakly normal subgroup of , and a nilpotent group, then is a weakly closed subgroup of .