# Sylow-relatively weakly closed subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

Suppose $P$ is a group of prime power order and $H$ is a subgroup of $P$. $H$ is a Sylow-relatively weakly closed subgroup of $P$ if, whenever $P$ is a Sylow subgroup of a finite group $G$, $H$ is a weakly closed subgroup of $P$ relative to $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Fusion system-relatively weakly closed subgroup weakly closed subgroup for any (saturated) fusion system on the whole group (obvious) |FULL LIST, MORE INFO
isomorph-normal coprime automorphism-invariant subgroup of group of prime power order the group is a group of prime power orderand the subgroup is both an isomorph-normal subgroup and a coprime automorphism-invariant subgroup (via fusion system-relatively weakly closed) |FULL LIST, MORE INFO
isomorph-normal characteristic subgroup of group of prime power order the group is a group of prime power orderand the subgroup is both an isomorph-normal subgroup and a characteristic subgroup (via isomorph-normal coprime automorphism-invariant subgroup) Fusion system-relatively weakly closed subgroup, Isomorph-normal coprime automorphism-invariant subgroup of group of prime power order|FULL LIST, MORE INFO
isomorph-free subgroup of group of prime power order the group is a group of prime power order and the subgroup is an isomorph-free subgroup (via isomorph-normal characteristic subgroup) Fusion system-relatively weakly closed subgroup, Isomorph-normal characteristic subgroup of group of prime power order, Isomorph-normal coprime automorphism-invariant subgroup of group of prime power order|FULL LIST, MORE INFO
characteristic maximal subgroup of group of prime power order the group is a group of prime power order, and the subgroup is both a characteristic subgroup and a maximal subgroup, and in particular has prime index (via isomorph-normal characteristic) Coprime automorphism-invariant maximal subgroup of group of prime power order, Fusion system-relatively weakly closed subgroup, Isomorph-normal characteristic subgroup of group of prime power order|FULL LIST, MORE INFO
Weaker than:fusion system-relatively strongly closed subgroup strongly closed subgroup for any (saturated) fusion system on the whole group (via fusion system-relatively weakly closed) Fusion system-relatively weakly closed subgroup|FULL LIST, MORE INFO
Sylow-relatively strongly closed subgroup strongly closed subgroup wherever the whole group is a Sylow subgroup of a bigger group |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions