Intermediately 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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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

Weaker properties

• Characteristic subgroup
• Potentially characteristic subgroup
• Normal subgroup

Related properties

• Image-closed characteristic subgroup

Metaproperties

Transitivity

It is not clear whether an intermediately characteristic subgroup of an intermediately characteristic subgroup is intermediately characteristic. The problem lies in this: if $H\le K\le G$, then there may be subgroups $M$ of $G$ which neither contain $K$ nor are contained in $K$.

Property operators

Right transiter

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

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