Idempotent subgroup property modifier

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

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.