Multi-invariance property

Definition
A subgroup property is termed a multi-invariance property if there exists, for every natural number $$n$$, a collection of functions $$F_n$$ from $$G^n$$ to $$G$$, such that a subgroup $$H$$ of $$G$$ satisfies the subgroup property iff the following holds:

$$\forall n, \forall f \in F_n, f(h_1,h_2,\ldots,h_n) \in H \ \forall \ h_1,h_2,\ldots,h_n \in H$$

A special case of this is an invariance property where the $$F_n$$ are empty for $$n > 1$$.

Stronger metaproperties

 * Invariance property

Weaker metaproperties

 * Strongly intersection-closed subgroup property:
 * Identity-true subgroup property
 * Intersection-closed subgroup property
 * ACU-closed subgroup property: