# Finite normal series operator

*This article defines a group property modifier (a unary group property operator) -- viz an operator that takes as input a group property and outputs a group property*

## Contents

## Definition

### Property-theoretic definition

The **finite normal series operator** is a group property modifier that takes as input a group property and outputs the Kleene star closure of with respect to the group extension operator, *right-bracketed*.

### Definition with symbols

Given a group property , the **finite normal series operator** applied to returns the group property defined as follows: a group has property if there is a finite normal series such that each is normal in and such that satisfies property for every .

## Relation with other modifiers

### Stronger modifiers

### Weaker modifiers

## Metaproperties

### Monotonicity

*This group property modifier is monotone, viz if are group properties and is the operator, then *

If are group properties, then the image of the finite normal series operator on implies the image of the finite normal series operator on .

### Ascendance

*This group property modifier is ascendant, viz the image of any group property under this modifier is always weaker than the group property we started with*

For any group property , any group satisfying also satisfies the image of under the finite normal series operator.

### Idempotence

The finite normal series operator need not be idempotent.