Intersect-all operator

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Definition

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: