Finite-intersection-closed subgroup property

Symbol-free definition
A subgroup property $$p$$ is termed finite-intersection-closed if an intersection of a finite nonempty collection of subgroups, each with property $$p$$, also has property $$p$$.

Definition with symbols
A subgroup property $$p$$ is termed finite-intersection-closed if whenever $$H_1, H_2, ..., H_n$$ are subgroups of $$G$$, each satisfying $$p$$ in $$G$$, then the intersection of all the $$H_i$$s also satisfies $$p$$ in $$G$$.

In terms of the intersection operator
The binary intersection operator takes two subgroup properties and gives the property of being a subgroup obtained as the intersection of subgroups with these properties. The intersection operator is a commutative associative quantalic binary operator, and the property of being finite-intersection-closed is the property of being transitive with respect to this operator.

A slight change in definition
We define being strongly finite-intersection-closed as being both finite-intersection-closed and identity-true. This is the same as being t.i. with respect to the intersection operator.

Stronger metaproperties

 * Strongly intersection-closed subgroup property
 * Intersection-closed subgroup property
 * Strongly finite-intersection-closed subgroup property
 * Transitive subgroup property satisfying transfer condition: