Intersect-all operator

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:

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