# Finite characteristic 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

### Definition with symbols

The **finite characteristic series operator** is a group property modifier that takes as input a group property and outputs a group property such that a group satisfies if and only if it has a characteristic series such that each quotient satisfies property .

## Relation with other modifiers

### Weaker modifiers

## Properties

### Monotonicity

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

### 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*

### Idempotence

*This group property modifier is idempotent, viz applying it twice to a group property has the same effect as applying it once*