Izable subgroup property

Definition with symbols
A subgroup property $$p$$ is termed izable if for any subgroup $$H$$ of a group $$G$$, there is a unique subgroup $$M$$ of $$G$$ such that all these conditions are satisfied:


 * $$M$$ contains $$H$$
 * Let $$L$$ be a subgroup of $$G$$ containing $$H$$. Then $$H$$ satisfies $$p$$ in $$L$$ if and only if $$L$$ is contained in $$M$$.

Equivalently, a subgroup property is izable if it is upper join-closed, identity-true and satisfies the intermediate subgroup condition.

Weaker metaproperties

 * Upper join-closed subgroup property
 * Identity-true subgroup property
 * Intermediate subgroup condition