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

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 $K$.

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
• Strongly potentially characteristic subgroup
• Potentially verbal subgroup
• Potentially fully characteristic subgroup
• Amalgam-characteristic subgroup
• Finite normal subgroup: For full proof, refer: Finite normal implies potentially characteristic
• Central subgroup: For full proof, refer: Central implies potentially characteristic
• Subgroup contained in a member of the upper central series

Weaker properties

• Potentially relatively characteristic subgroup
• Normal subgroup: For full proof, refer: Potentially characteristic implies normal

Conjecture of equalling normality

This property is conjectured to equal the property: normality

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. However, it is true that any finite normal subgroup 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 normal implies potentially characteristic,Central implies 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 $H$.