# Intersect-all operator

From Groupprops

## 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 be a subgroup property. The intersect-all operator applied to outputs the following subgroup-defining function :

where varies over all subgroups of satisfying property in .

## Application

Important instances of application of the intersect-all operator:

- Frattini subgroup: obtained from maximal subgroup
- Jacobson radical: obtained from maximal normal subgroup