Intermediately 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]

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Definition

Symbol-free definition

A subgroup of a group is said to be intermediately characteristic if it is characteristic not only in the whole group but also in every intermediate subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is said to be intermediately characteristic if forany intermediate subgroup $K$ (such that $H\le K\le G$), $H$ is characteristic in $K$.

In terms of the intermediately operator

The subgroup property of being intermediately characteristic can be obtained by applying the intermediately operator to the subgroup property of being characteristic.

Relation with other properties

Stronger properties

• Isomorph-free subgroup
• Order-unique subgroup
• Transfer-closed characteristic subgroup: For proof of the implication, refer Transfer-closed characteristic implies intermediately characteristic and for proof of its strictness (i.e. the reverse implication being false) refer intermediately characteristic not implies transfer-closed characteristic.

Weaker properties

• Characteristic subgroup
• Potentially characteristic subgroup
• Normal subgroup

Related properties

• Image-closed characteristic subgroup

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

For full proof, refer: Intermediate characteristicity is not transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

For full proof, refer: Intermediate characteristicity is quotient-transitive

Effect of property operators

Right transiter

It turns out that any intermediately characteristic subgroup of a transfer-closed characteristic subgroup is again intermediately characteristic. This follows from some simple reasoning and the fact that characteristicity is itself transitive. Further information: Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic

Hence, the right transiter of the property of being intermediately characteristic is weaker than the property of being transfer-closed characteristic.