Characteristic-potentially characteristic subgroup: Difference between revisions

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]
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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
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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 $H$ of a group $G$ is termed characteristic-potentially characteristic in $G$ if there exists a group $K$ containing $G$ such that both $H$ and $G$ are characteristic in $K$ .

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 $p$ and $q$ , the upper-hook operator of $p$ and $q$ is defined as the following property $r$ : a subgroup $H$ of a group $K$ has property $r$ if there exists a group $G$ containing $K$ such that $H$ has property $p$ in $G$ and $K$ has property $q$ in $G$ .

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

Characteristic subgroup

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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@@ Line 1: / Line 1: @@
+{{wikilocal}}
 {{subgroup property}}
+{{variation of|characteristicity}}
+{{termrelatedto|NPC conjecture}}
+==Definition==
-{{wikilocal}}
+===Symbol-free definition===
-{{variationof|characteristicity}}
+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==
+===Definition with symbols===
-===Symbol-free definition===
+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>.
-A subgroup of a group is termed '''strongly 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.
+==Formalisms==
-===Definition with symbols===
+{{wikilocal-section}}
-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 both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.
+===In terms of the upper-hook operator===
-===In terms of the strongly potentially 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 [[subgroup property]] of being '''potentially characteristic''' is obtained by applying the [[strongly potentially operator]] to the subgroup property of being [[characteristic subgroup|characteristic]].
+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==
@@ Line 23: / Line 27: @@
 ===Stronger properties===
-* [[Characteristic subgroup]]
+* [[Weaker than::Characteristic subgroup]]
 ===Weaker properties===
-* [[Potentially characteristic subgroup]]
+* [[Stronger than::Normal-potentially characteristic subgroup]]
-* [[Potentially relatively characteristic subgroup]]
+* [[Stronger than::Normal-potentially relatively characteristic subgroup]]
-* [[Extensible automorphism-invariant subgroup]]
+* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat|[[Characteristic-potentially characteristic implies normal]]|[[Normal not implies characteristic-potentially characteristic]]}}
-* [[Normal subgroup]]
+* [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]
+* [[Stronger than::Normal-extensible automorphism-invariant subgroup]]
-===Conjecture of equalling normality===
-{conjecturedtoequal|normality}}
-The [[NSPC conjecture]] states that every normal subgorup is strongly potentially characteristic. In other words, if <math>H \triangleleft G</math>, there is a group <math>K</math> containing <math>G</math> such that both both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.
 ==Metaproperties==
@@ Line 47: / Line 46: @@
 {{fillin}}
-==Property operators==
-===Left transiter===
-Every characteristic subgroup of a strongly potentially characteristic subgroup is strongly potentially characteristic. In fact, the same supergroup works.
-That is, suppose <math>M \le G \le H</math> with <math>M</math> characteristic in <math>G</math> and <math>G</math> strongly 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 both <math>M</math> and <math>H</math> are characteristic in <math>K</math>, and hence <math>M</math> is strongly potentially characteristic in <math>H</math>.