Intersect-all operator

Symbol-free definition
The intersect-all operator is an operator that takes as input a subgroup property and outputs the subgroup-defining function that sends a group to the intersection of all subgroups of it that satisfy the property.

Definition with symbols
Let $$p$$ be a subgroup property. The intersect-all operator applied to $$p$$ outputs the following subgroup-defining function $$f$$:

$$f(G) = \bigcap H$$

where $$H$$ varies over all subgroups of $$G$$ satisfying property $$p$$ in $$G$$.

Application
Important instances of application of the intersect-all operator: