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| {{subgroup property}}
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| {{wikilocal}}
| | * [[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. |
| {{variationof|characteristicity}}
| | * [[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. |
| ==History==
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| {{wikilocal-see-talk-page}}
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| ==Definition==
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| ===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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| {{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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| * [[Characteristic subgroup]] | |
| * [[Intermediately characteristic subgroup]]
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| * [[Strongly potentially characteristic subgroup]] | |
| * [[Potentially fully characteristic subgroup]]
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| ===Weaker properties===
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| * [[Potentially relatively characteristic subgroup]]
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| * [[Extensible automorphism-invariant subgroup]]
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| * [[Normal subgroup]]
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| ===Conjecture of equalling normality===
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| {{conjecturedtoequal|normality}}
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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. This is discussed at the page on the [[NPC conjecture]].
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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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| 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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