Sylow-relatively weakly closed subgroup

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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.