Strongly intersection-closed subgroup property

Symbol-free definition
A subgroup property is said to be strongly intersection-closed if given any arbitrary (possibly empty) family of subgroups each satisfying the subgroup property in the group, the intersection of all the subgroups again satisfies the property.

Definition with symbols
A subgroup property $$p$$ is termed strongly intersection-closed if given any group $$G$$ and any (possibly empty) family of subgroups $$H_i$$ of $$G$$ indexed by $$i \in I$$ such that each $$H_i$$ satisfies $$p$$ in $$G$$, the group $$\bigcap_{i \in I} H_i$$ also satisfies $$p$$ in $$G$$.

In other words, a subgroup property is strongly intersection-closed if it is both intersection-closed and identity-true.

Stronger metaproperties

 * Weaker than::Strongly UL-intersection-closed subgroup property
 * Weaker than::Invariance property:

Weaker metaproperties

 * Stronger than::Intersection-closed subgroup property
 * Stronger than::Finite-intersection-closed subgroup property
 * Stronger than::Identity-true subgroup property