Idempotent subgroup-defining function

This article defines a property of subgroup-defining functions, viz., a property that any subgroup-defining function may either satisfy or not satisfy

This article defines a notion of an idempotent (one that equals its square) in a certain context

Definition

Definition with symbols

A subgroup-defining function $f$ is said to be idempotent if for any group $G$, $f(f(G))=f(G)$ (that is, they both refer to the same subgroup of $G$).

Subgroup-defining functions satisfying this property

A full listing is available at:

Center

The center of the center of a group is again the center.

Any group that arises as the center of some group must be Abelian, and any Abelian group is its own center.