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| {{wikilocal}}
| | You might be looking for: |
| {{subgroup property}}
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| {{variationof|characteristicity}}
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| {{termrelatedto|NPC conjecture}}
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| ==History==
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| {{wikilocal-see-talk-page}}
| | * [[normal equals potentially characteristic]]: A [[normal subgroup]] <math>H</math> of a [[group]] <math>G</math> is potentially characteristic: there exists a group <math>K</math> containing <math>G</math> such that <math>H</math> is a [[characteristic subgroup]] of <math>K</math>. |
| | | * [[group property-conditionally potentially characteristic subgroup]]: This is the notion of being potentially characteristic with respect to a group property. |
| ==Definition==
| | * [[potentially characteristic subgroups characterization problem]] |
| | | * [[potentially operator]]: This takes as input a subgroup property and outputs the property of being a subgroup that can satisfy the property in some ambient group. |
| ===Symbol-free definition===
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| A subgroup of a group is termed '''potentially characteristic''' if there is an embedding of the bigger group in some group such that, in that embedding the subgroup becomes [[characteristic subgroup|characteristic]].
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| ===Definition with symbols===
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| A subgroup <math>H</math> of a group <math>G</math> is termed '''potentially characteristic''' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that <math>H</math> is [[characteristic subgroup|characteristic]] in <math>K</math>.
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| ==Formalisms==
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| {{obtainedbyapplyingthe|potentially operator|characteristic subgroup}}
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| The property of being potentially characteristic is obtained by applying the [[potentially operator]] to the property of being [[characteristic subgroup|characteristic]]. The potentially operator is an idempotent ascendant monotone operator.
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| ==Relation with other properties==
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| ===Stronger properties===
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| * [[Weaker than::Characteristic subgroup]]
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| * [[Weaker than::Intermediately characteristic subgroup]]
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| * [[Weaker than::Characteristic-potentially characteristic subgroup]]
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| * [[Weaker than::Normal-potentially characteristic subgroup]]
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| * [[Weaker than::Potentially verbal subgroup]]
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| * [[Weaker than::Potentially fully invariant subgroup]]
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| * [[Weaker than::Retract-potentially characteristic subgroup]]
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| * [[Weaker than::Amalgam-characteristic subgroup]]
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| ** [[Weaker than::Finite normal subgroup]]: {{proofat|[[Finite normal implies potentially characteristic]]}}
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| ** [[Weaker than::Central subgroup]]: {{proofat|[[Central implies potentially characteristic]]}}
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| ** [[Weaker than::Normal subgroup contained in the hypercenter]]: {{proofat|[[Normal subgroup contained in hypercenter is potentially characteristic]]}}
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| ===Weaker properties===
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| * [[Stronger than::Normal subgroup]]: {{proofat|[[Potentially characteristic implies normal]]}}
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| * Potentially relatively characteristic subgroup: There is an ''a priori'' weaker definition of potentially ''relatively'' characteristic subgroup. However, this weaker definition turns out to be equal to normality. Nonetheless, the notion is distinct from normality when working with subvarieties of the variety of groups. {{proofat|[[Potentially relatively characteristic equals normal]]}}
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| ===Related properties===
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| * [[Finite-potentially characteristic subgroup]] | |
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| ===Conjecture of equalling normality===
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| {{conjecturedtoequal|normality}}
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| {{further|[[NPC conjecture]]}}
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| Since the potentially operator is an idempotent monotone ascendant operator, and the property of being [[normal subgroup|normal]] is a fixed point of this operator, every potentially characteristic subgroup is normal. The converse question: ''is every normal subgroup potentially characteristic?'' has not yet been answered.
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| The statement is true for finite groups. Also, it is true that any [[finite normal subgroup]] of a group is potentially characteristic, and it is also true that any normal subgroup of a [[nilpotent group]] (and more generally, any normal subgroup contained in a member of the [[upper central series]]) is potentially characteristic. {{proofat|[[Finite NPC theorem]], [[Finite normal implies potentially characteristic]],[[Central implies potentially characteristic]], [[Normal subgroup contained in hypercenter is potentially characteristic]]}}
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| ==Metaproperties==
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| {{intransitive}}
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| {{fillin}}
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| {{intersection-closed-open}}
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| Is the intersection of two potentially characteristic subgroups potentially characteristic?
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| ==Property operators==
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| ===Left transiter===
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| {{further|[[Characteristic of potentially characteristic implies potentially characteristic]]}}
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| Every characteristic subgroup of a potentially characteristic subgroup is potentially characteristic. In fact, the same supergroup works.
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| That is, suppose <math>M \le G \le H</math> with <math>M</math> characteristic in <math>G</math> and <math>G</math> potentially characteristic in <math>H</math>. Then, there exists a group <math>K</math> containing <math>H</math> such that both <math>G</math> and <math>H</math> are [[characteristic subgroup|characteristic]] in <math>K</math>. Then, we also have that <math>M</math> is characteristic in <math>K</math>, and hence <math>M</math> is potentially characteristic in <math>H</math>.
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