Potentially characteristic subgroup

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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
View other terms related to NPC conjecture | View facts related to NPC conjecture

History

This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page

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 $H$ of a group $G$ is termed potentially characteristic in $G$ if there exists a group $K$ containing $G$ such that $H$ is characteristic in $K$.

Formalisms

In terms of the potentially operator

This property is obtained by applying the potentially operator to the property: characteristic subgroup
View other properties obtained by applying the potentially operator

The property of being potentially characteristic is obtained by applying the potentially operator to the property of being characteristic. The potentially operator is an idempotent ascendant monotone operator.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Intermediately characteristic subgroup
• Characteristic-potentially characteristic subgroup
• Normal-potentially characteristic subgroup
• Potentially verbal subgroup
• Potentially fully invariant subgroup
• Retract-potentially characteristic subgroup
• Amalgam-characteristic subgroup
  • Finite normal subgroup: For full proof, refer: Finite normal implies potentially characteristic
  • Periodic normal subgroup: For full proof, refer: Periodic normal implies potentially characteristic
  • Central subgroup: For full proof, refer: Central implies potentially characteristic
  • Normal subgroup contained in the hypercenter: For full proof, refer: Normal subgroup contained in hypercenter is potentially characteristic

Weaker properties

• Normal subgroup: For full proof, refer: Potentially characteristic implies normal
• Potentially relatively characteristic subgroup: There is an a priori weaker definition of potentially relatively characteristic subgroup. However, this weaker definition turns out to be equal to normality. Nonetheless, the notion is distinct from normality when working with subvarieties of the variety of groups. For full proof, refer: Potentially relatively characteristic equals normal

Related properties

• Finite-potentially characteristic subgroup

Conjecture of equalling normality

This property is conjectured to equal the property: normality

Further information: NPC conjecture

Since the potentially operator is an idempotent monotone ascendant operator, and the property of being 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.

The statement is true for finite groups. Also, it is true that any finite normal subgroup of a group is potentially characteristic, and it is also true that any normal subgroup of a nilpotent group (and more generally, any normal subgroup contained in a member of the upper central series) is potentially characteristic. For full proof, refer: Finite NPC theorem, Finite normal implies potentially characteristic,Central implies potentially characteristic, Normal subgroup contained in hypercenter is potentially characteristic

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.

Is the intersection of two potentially characteristic subgroups potentially characteristic?

Property operators

Left transiter

Further information: Characteristic of potentially characteristic implies potentially characteristic

Every characteristic subgroup of a potentially characteristic subgroup is potentially characteristic. In fact, the same supergroup works.

That is, suppose $M\le G\le H$ with $M$ characteristic in $G$ and $G$ potentially characteristic in $H$. Then, there exists a group $K$ containing $H$ such that both $G$ and $H$ are characteristic in $K$. Then, we also have that $M$ is characteristic in $K$, and hence $M$ is potentially characteristic in $H$.