Potentially fully invariant subgroup

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This is a variation of full characteristicity|Find other variations of full characteristicity |

Definition

Symbol-free definition

A subgroup of a group is termed potentially fully characteristic if there is an embedding of the bigger group in some group such that, in that embedding the subgroup becomes fully characteristic.

Definition with symbols

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

In terms of the potentially operator

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

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

Relation with other properties

Stronger properties

• Fully characteristic subgroup
• Strongly potentially fully characteristic subgroup

Weaker properties

• Potentially 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 fully characteristic subgroup is normal. The converse question: is every normal subgroup potentially fully characteristic? has not yet been answered.

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
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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

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

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