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

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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.

Relation with other properties

Stronger properties

Weaker properties

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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