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]
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This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

History

Origin

The notion of potentially characteristic subgroup has been introduced by Vipul Naik. So far, it is only local to the wiki.

The notion of potentially characteristic subgroup was introduced by Vipul Naik to study ideas that came up naturally while working on the Extensible Automorphisms Problem.

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 $K$ .

Definition in terms of 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

Weaker properties

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

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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?