Closure-characteristic subgroup: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::Characteristic subgroup]] || invariant under all [[automorphism]]s || || || {{intermediate notions short|closure-characteristic subgroup|characteristic subgroup}} | |||
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| [[Weaker than::Automorph-conjugate subgroup]] || all [[automorphic subgroups]] are [[conjugate subgroups|conjugate]] || || || {{intermediate notions short|closure-characteristic subgroup|automorph-conjugate subgroup}} | |||
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| [[Weaker than::Join of automorph-conjugate subgroups]] || [[join of subgroups|join]] of [[automorph-conjugate subgroups]] || || || {{intermediate notions short|closure-characteristic subgroup|join of automorph-conjugate subgroups}} | |||
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| [[Weaker than::Sylow subgroup]] || <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> || || || {{intermediate notions short|closure-characteristic subgroup|Sylow subgroup}} | |||
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| [[Weaker than::Hall subgroup]] || subgroup of finite group whose order and index are relatively prime || || || {{intermediate notions short|closure-characteristic subgroup|Hall subgroup}} | |||
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| [[Weaker than::Contranormal subgroup]] || [[normal closure]] is whole group || || || {{intermediate notions short|closure-characteristic subgroup|contranormal subgroup}} | |||
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===Conjunction with other properties=== | ===Conjunction with other properties=== | ||
Revision as of 23:47, 11 January 2010
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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
QUICK PHRASES: normal closure is characteristic, join of all conjugates is characteristic
Symbol-free definition
A subgroup of a group is termed closure-characteristic if its normal closure in the whole group is a characteristic subgroup.
Definition with symbols
A subgroup of a group is termed closure-characteristic if the normal closure of in is characteristic in .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Characteristic subgroup | invariant under all automorphisms | |FULL LIST, MORE INFO | ||
| Automorph-conjugate subgroup | all automorphic subgroups are conjugate | |FULL LIST, MORE INFO | ||
| Join of automorph-conjugate subgroups | join of automorph-conjugate subgroups | |FULL LIST, MORE INFO | ||
| Sylow subgroup | -subgroup of finite group with index relatively prime to | |FULL LIST, MORE INFO | ||
| Hall subgroup | subgroup of finite group whose order and index are relatively prime | |FULL LIST, MORE INFO | ||
| Contranormal subgroup | normal closure is whole group | |FULL LIST, MORE INFO |
Conjunction with other properties
Any normal subgroup that is also closure-characteristic, is characteristic.
Metaproperties
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
Join-closedness
YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness