Idempotent subgroup-defining function

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:

Category:Idempotent subgroup-defining functions

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.