Intermediately characteristic subgroup

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 $$K$$.

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

Stronger properties

 * Weaker than::Isomorph-free subgroup
 * Weaker than::Order-unique subgroup
 * Weaker than::Transfer-closed characteristic subgroup:

Weaker properties

 * Stronger than::Characteristic subgroup
 * Stronger than::Potentially characteristic subgroup
 * Stronger than::Normal subgroup

Related properties

 * Image-closed characteristic subgroup

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.

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