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 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
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.