Characteristic-potentially characteristic subgroup: Difference between revisions
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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>. | 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 | The property of being characteristic-potentially characteristic is thus obtained by applying the upper-hook operator to the property [[characteristic subgroup]] with itself. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 04:59, 2 May 2022
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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]
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity
This term is related to: NPC conjecture
View other terms related to NPC conjecture | View facts related to 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 of a group is termed characteristic-potentially characteristic in if there exists a group containing such that both and are characteristic in .
Formalisms
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
In terms of the upper-hook operator
Given two subgroup properties and , the upper-hook operator of and is defined as the following property : a subgroup of a group has property if there exists a group containing such that has property in and has property in .
The property of being characteristic-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 properties
- Normal-potentially characteristic subgroup
- Normal-potentially relatively characteristic subgroup
- Normal subgroup: For proof of the implication, refer Characteristic-potentially characteristic implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies characteristic-potentially characteristic.
- Characteristic-extensible automorphism-invariant subgroup
- Normal-extensible automorphism-invariant subgroup
Metaproperties
Transitivity
NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity
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Intersection-closedness
The problem of whether an intersection (finite or arbitrary) of subgroups with this property again has this property is an open problem.
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