# Poly 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 **poly operator** takes as input a group property and outputs the Kleene star closure of with respect to the extension operator, bracketed on the left.

### Definition with symbols

Given a group property , the poly operator gives the group property defined as follows:

A group has property if we can find a subnormal series such that each satisfies property as an abstract group.

## Relation with other modifiers

### Stronger property modifiers

## Properties

### Monotonicity

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

If are group properties, 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*

For any group property , . In other words, if a group satisfies property it also satisfies property .

### Idempotence

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

For any group property , . In other words, applying the poly operator twice has the same effect as applying it once.