Idempotent subgroup property modifier

Definition
A subgroup property modifier $$F$$ is termed idempotent if it satisfies the following equivalent conditions:


 * $$F^2 = F$$
 * The fixed-point space of $$F$$ (in other words, those subgroup properties that are unchanged by $$F$$) coincides with the image space of $$F$$ (in other words, those subgroup properties that arise by applying $$F$$ to some subgroup property)

The notion of idempotence is general -- one can talk of an idempotent property modifier.