Potentially characteristic subgroup

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Potentially characteristic subgroup, all facts related to Potentially characteristic subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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 fully characteristic subgroup

Weaker properties

• Potentially relatively characteristic subgroup
• Extensible automorphism-invariant subgroup
• Normal subgroup

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. This is discussed at the page on the NPC conjecture.

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

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

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

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 $H$.