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]
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
|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|
- 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.