Intermediately operator

Symbol-free definition
The intermediately operator is a map from the subgroup property space to itself, that sends a subgroup property $$p$$ to the property of being a subgroup that satisfies $$p$$ not only in the whole group, but also in every intermediate subgroup.

Definition with symbols
Given a subgroup property $$p$$, the subgroup property intermediately $$p$$ is the property as follows: $$H$$ satisfies intermediately $$p$$ in $$G$$ if for any group $$K$$ with $$H \le K \le G$$, $$H$$ satisfies $$p$$ in $$K$$.

Properties
If $$p \le q$$ are two subgroup properties, then intermediately $$p \le$$ intermediately $$q$$. This follows directly from the definition.

For any subgroup property $$p$$, intermediately $$p \le p$$. This follows from the fact that if $$H$$ satisfies property $$p$$ in every intermediate subgroup, $$H$$ also satisfies property $$p$$ in the whole group.

The intermediately operator is idempotent, in the sense that applying the intermedaitely operator twice has the same effect as applying it once. The image-cum-fixed-point-space for this operator is precisely the subgroup properties satisfying the intermediate subgroup condition.

Effect on metaproperties
Suppose $$p$$ is a join-closed subgroup property, viz the join of any family of subgroups satisfying property $$p$$, also satisfies property $$p$$. Then, it is easy to see that intermediately $$p$$ is also join-closed.

Transitivity
It is not clear whether, even if $$p$$ is transitive, intermediately $$p$$ will b transitive.

Transfer condition
If $$p$$ satisfies the transfer condition, it also, in particular, satisfies the intermediate subgroup condition, and hence $$p$$ is unchanged under application of the intermediately operator.

Naturally arising properties that satisfy intermediate subgroup condition
These include:


 * Normal subgroup
 * Central factor
 * Conjugacy-closed subgroup
 * Permutable subgroup
 * Modular subgroup

Subgroup properties obtained by explicitly applying the intermediately operator

 * Intermediately characteristic subgroup
 * Compressed subgroup