Core-characteristic subgroup: Difference between revisions
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Weaker than:: | | [[Weaker than::characteristic subgroup]] || invariant under all [[automorphism]]s|| || || {{intermediate notions short|core-characteristic subgroup|characteristic subgroup}} | ||
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| [[Weaker than:: | | [[Weaker than::automorph-dominating sugroup]] || every [[automorphic subgroups|automorphic subgroup]] is contained in a [[conjugate subgroups|conjugate subgroup]] || || || {{intermediate notions shortcore-characteristic subgroup|automorph-dominating subgroup}} | ||
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| [[Weaker than:: | | [[Weaker than::automorph-conjugate subgroup]] || every [[automorphic subgroups|automorphic subgroup]] is [[conjugate subgroups|conjugate]] to it || (via automorph-dominating) || || {{intermediate notions short|core-characteristic subgroup|automorph-conjugate subgroup}} | ||
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| [[Weaker than:: | | [[Weaker than::intersection of automorph-conjugate subgroups]] || intersection of [[automorph-conjugate subgroup]]s || || || {{intermediate notions short|core-characteristic subgroup|intersection of automorph-conjugate subgroups}} | ||
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| [[Weaker than::core-free subgroup]] || [[normal core]] is trivial || || || {{intermediate notions short|core-characteristic subgroup|core-free subgroup}} | |||
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| [[Weaker than::Sylow subgroup]] || <math>p</math>-subgroup of finite group whose index is relatively prime to <math>p</math> || || || {{intermediate notions short|core-characteristic subgroup|Sylow subgroup}} | | [[Weaker than::Sylow subgroup]] || <math>p</math>-subgroup of finite group whose index is relatively prime to <math>p</math> || || || {{intermediate notions short|core-characteristic subgroup|Sylow subgroup}} | ||
Revision as of 17:34, 21 December 2014
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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
QUICK PHRASES: intersection of all conjugates is characteristic, normal core is characteristic
Symbol-free definition
A subgroup of a group is termed core-characteristic if its normal core is a characteristic subgroup of the whole group.
Definition with symbols
A subgroup of a group is termed core-characteristic if the normal core of in is a characteristic subgroup of .
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-dominating sugroup | every automorphic subgroup is contained in a conjugate subgroup | Template:Intermediate notions shortcore-characteristic subgroup | ||
| automorph-conjugate subgroup | every automorphic subgroup is conjugate to it | (via automorph-dominating) | |FULL LIST, MORE INFO | |
| intersection of automorph-conjugate subgroups | intersection of automorph-conjugate subgroups | |FULL LIST, MORE INFO | ||
| core-free subgroup | normal core is trivial | |FULL LIST, MORE INFO | ||
| Sylow subgroup | -subgroup of finite group whose index is relatively prime to | |FULL LIST, MORE INFO | ||
| Hall subgroup | subgroup of finite group whose order and index are relatively prime | |FULL LIST, MORE INFO |
Conjunction with other properties
Any normal subgroup that is also core-characteristic, is characteristic.