Characteristic-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]
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

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

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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 strongly potentially characteristic is thus obtained by applying the upper-hook operator to the property characteristic subgroup with itself.

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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