Potentially characteristic subgroup: Difference between revisions

Latest revision as of 07:00, 22 February 2013

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normal equals potentially characteristic: A normal subgroup $H$ of a group $G$ is potentially characteristic: there exists a group $K$ containing $G$ such that $H$ is a characteristic subgroup of $K$ .
group property-conditionally potentially characteristic subgroup: This is the notion of being potentially characteristic with respect to a group property.
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.

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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==
-===Origin===
-The notion of potentially characteristic subgroup has been introduced by [[user:Vipul|Vipul Naik]]. So far, it is only local to the wiki.
-The notion of '''potentially characteristic subgroup''' was introduced by Vipul Naik to study ideas that came up naturally while working on the [[Extensible Automorphisms Problem]].
-==Definition==
-===Symbol-free definition===
-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.
-===Definition with symbols===
-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 in <math>K</math>.
-===Definition in terms of the potentially operator===
-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.
-==Relation with other properties==
-===Stronger properties===
-* [[Characteristic subgroup]]
-* [[Intermediately characteristic subgroup]]
-* [[Strongly potentially characteristic subgroup]]
-===Weaker properties===
-* [[Normal subgroup]]
-===Conjecture of equalling normality===
-{{conjecturedtoequal|normality}}
-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]].
-==Metaproperties==
-{{intransitive}}
-{{fillin}}
-{{intersection-closed-open}}
-Is the intersection of two potentially characteristic subgroups potentially characteristic?