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.