# Idempotent subgroup-defining function

From Groupprops

*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*

## Contents

## Definition

### Definition with symbols

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

## Relation with other properties

## 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.