Sub-closure operator

Definition
Let $$p$$ be a strongly intersection-closed subgroup property, viz a subgroup property that is satisfied by the whole group as a subgroup of itself, and such that an arbitrary intersection of subgroups with property $$p$$ also satisfies property $$p$$.

Then, a subgroup $$H$$ of a group $$G$$ is said to be a $$n$$ times sub-$$p$$-closure if we have a sequence:

$$G = K_0 \ge K_1 \ge \ldots K_n = H$$

where $$K_i$$ is the smallest subgroup containing $$H$$ that satisfies property $$p$$ in $$K_{i-1}$$.

A subgroup is termed a sub-$$p$$-closure if it is a $$n$$ times sub-$$p$$-closure for some finite positive integer $$n$$.

In general
In general, we have three different things:


 * The subordination of the property, which is the weakest
 * The sub-closure of the property, which is somewhere in between
 * The property itself, which is the strongest.

When it is the same as the original property
Clearly, if the given subgroup property is transitive it equals both its sub-closure and its subordination.

When the sub-closure equals the subordination
Clearly, in the event that the subgroup property is itself transitive, it equals both. However, even if it is not transitive, it is common to expect that the sub-closure equals the subordination. This happens, for instance, if the subgroup property satisfies the intermediate subgroup condition.