View source for Characteristic-potentially characteristic subgroup

{{wikilocal}}
{{subgroup property}}
{{variation of|characteristicity}}
{{termrelatedto|NPC conjecture}}
==Definition==

===Symbol-free definition=== 

A subgroup of a group is termed '''characteristic-potentially characteristic''' if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.

===Definition with symbols===

A subgroup <math>H</math> of a group <math>G</math> is termed '''characteristic-potentially characteristic''' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.

==Formalisms==

{{wikilocal-section}}

===In terms of the upper-hook operator===

Given two subgroup properties <math>p</math> and <math>q</math>, the [[upper-hook operator]] of <math>p</math> and <math>q</math> is defined as the following property <math>r</math>: a subgroup <math>H</math> of a group <math>K</math> has property <math>r</math> if there exists a group <math>G</math> containing <math>K</math> such that <math>H</math> has property <math>p</math> in <math>G</math> and <math>K</math> has property <math>q</math> in <math>G</math>.

The property of being strongly potentially characteristic is thus obtained by applying the upper-hook operator to the property [[characteristic subgroup]] with itself.

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Characteristic subgroup]]

===Weaker properties===

* [[Stronger than::Normal-potentially characteristic subgroup]]
* [[Stronger than::Normal-potentially relatively characteristic subgroup]]
* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat|[[Strongly potentially characteristic implies potentially characteristic]]|[[Potentially characteristic not implies strongly potentially characteristic]]}}
* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat|[[Characteristic-potentially characteristic implies normal]]|[[Normal not implies characteristic-potentially characteristic]]}}
* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat|[[Potentially relatively characteristic equals normal]]}}
* [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]
* [[Stronger than::Normal-extensible automorphism-invariant subgroup]]

==Metaproperties==

{{intransitive}}

{{fillin}}

{{intersection-closed-open}}

{{fillin}}