# Sylow-relatively weakly closed subgroup

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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 is a group of prime power order and is a subgroup of . is a **Sylow-relatively weakly closed subgroup** of if, whenever is a Sylow subgroup of a finite group , is a weakly closed subgroup of relative to .

## Relation with other properties

### Stronger properties

### Weaker properties

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

Hall-relatively weakly closed subgroup | |FULL LIST, MORE INFO | |||

normal subgroup of group of prime power order, normal subgroup | Coprime automorphism-invariant normal subgroup of group of prime power order|FULL LIST, MORE INFO | |||

coprime automorphism-invariant normal subgroup of group of prime power order coprime automorphism-invariant normal subgroup | Sylow-relatively weakly closed implies coprime automorphism-invariant normal | |FULL LIST, MORE INFO |

### Incomparable properties

- Characteristic subgroup of group of prime power order: Note that both the property of being Sylow-relatively weakly closed and the property of being characteristic are qualitatively similar in that they describe a sort of invariance that is intermediate between being isomorph-free and being normal. To be more precise, they are both sandwiched between the property of being a coprime automorphism-invariant normal subgroup and the property of being an isomorph-normal characteristic subgroup.