Sylow-relatively weakly closed subgroup: Difference between revisions

Latest revision as of 03:10, 6 June 2015

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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	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)	\|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)	\|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)	\|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)	\|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	Proof of implication	Intermediate notions
Hall-relatively weakly closed subgroup		\|FULL LIST, MORE INFO
normal subgroup of group of prime power order, normal subgroup		\|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.

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 ===Stronger properties===
-* [[Weaker than::Isomorph-normal coprime automorphism-invariant subgroup of group of prime power order]]: {{proofat|[[Isomorph-normal coprime automorphism-invariant implies weakly closed for any fusion system]]}}
+{| class="sortable" border="1"
-** [[Weaker than::Isomorph-free subgroup of group of prime power order]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-** [[Weaker than::Characteristic maximal subgroup of group of prime power order]]
+|-
-** [[Weaker than::Isomorph-normal characteristic subgroup of group of prime power order]]
+| [[Weaker than::Fusion system-relatively weakly closed subgroup]] || [[weakly closed subgroup for a fusion system|weakly closed subgroup]] for any (saturated) fusion system on the whole group || (obvious) || || {{intermediate notions short|Sylow-relatively weakly closed subgroup|fusion system-relatively weakly closed subgroup}}
-* [[Weaker than::Fusion system-relatively strongly closed subgroup]]
+|-
-* [[Weaker than::Sylow-relatively strongly closed subgroup]]
+| [[Weaker than::isomorph-normal coprime automorphism-invariant subgroup of group of prime power order]] || the group is a [[group of prime power order]]and the subgroup is both an [[isomorph-normal subgroup]] and a [[coprime automorphism-invariant subgroup]] || ([[isomorph-normal coprime automorphism-invariant implies weakly closed for any fusion system|via fusion system-relatively weakly closed]]) || || {{intermediate notions short|Sylow-relatively weakly closed subgroup|isomorph-normal coprime automorphism-invariant subgroup}}
-* [[Weaker than::Fusion system-relatively weakly closed subgroup]]
+|-
+| [[Weaker than::isomorph-normal characteristic subgroup of group of prime power order]] || the group is a [[group of prime power order]]and the subgroup is both an [[isomorph-normal subgroup]] and a [[characteristic subgroup]] || (via isomorph-normal coprime automorphism-invariant subgroup) || || {{intermediate notions short|Sylow-relatively weakly closed subgroup|isomorph-normal characteristic subgroup of group of prime power order}}
+|-
+| [[Weaker than::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) || || {{intermediate notions short|Sylow-relatively weakly closed subgroup|isomorph-free subgroup of group of prime power order}}
+|-
+| [[Weaker than::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 of group of prime power order|maximal subgroup]], and in particular has prime index || (via isomorph-normal characteristic) || || {{intermediate notions short|Sylow-relatively weakly closed subgroup|characteristic maximal subgroup of group of prime power order}}
+|-
+| [[Weaker than:fusion system-relatively strongly closed subgroup]] || [[strongly closed subgroup for a fusion system|strongly closed subgroup]] for any (saturated) fusion system on the whole group || (via fusion system-relatively weakly closed) || || {{intermediate notions short|Sylow-relatively weakly closed subgroup|fusion system-relatively strongly closed subgroup}}
+|-
+| [[Weaker than::Sylow-relatively strongly closed subgroup]] || [[strongly closed subgroup]] wherever the whole group is a [[Sylow subgroup]] of a bigger group || || || {{intermediate notions short|Sylow-relatively weakly closed subgroup|Sylow-relatively strongly closed subgroup}}
+|}
+===Weaker properties===
-===Weaker properties===
-* [[Stronger than::Hall-relatively weakly closed subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::Normal subgroup of group of prime power order]], [[stronger than::normal subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Coprime automorphism-invariant normal subgroup of group of prime power order]], [[stronger than::coprime automorphism-invariant normal subgroup]]: {{proofat|[[Sylow-relatively weakly closed implies coprime automorphism-invariant normal]]}}
+|-
+| [[Stronger than::Hall-relatively weakly closed subgroup]] || || || || {{intermediate notions short|Hall-relatively weakly closed subgroup|Sylow-relatively weakly closed subgroup}}
+|-
+| [[Stronger than::normal subgroup of group of prime power order]], [[Stronger than::normal subgroup]] || || || || {{intermediate notions short|normal subgroup of group of prime power order|Sylow-relatively weakly closed subgroup}}
+|-
+| [[Stronger than::coprime automorphism-invariant normal subgroup of group of prime power order]] [[stronger than::coprime automorphism-invariant normal subgroup]] || || [[Sylow-relatively weakly closed implies coprime automorphism-invariant normal]] || || {{intermediate notions short|coprime automorphism-invariant normal subgroup of group of prime power order|Sylow-relatively weakly closed subgroup}}
+|}
 ===Incomparable properties===
-* [[Characteristic subgroup of group of prime power order]]
+* [[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.