# 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

## Definition

### Property-theoretic definition

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

### Definition with symbols

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

A group $G$ has property $q$ if we can find a subnormal series $e = H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_r = G$ such that each $H_r/H_{r-1}$ satisfies property $p$ as an abstract group.

## Properties

### Monotonicity

This group property modifier is monotone, viz if $p \le q$ are group properties and $f$ is the operator, then $f(p) \le f(q)$

If $p \le q$ are group properties, then $poly(p) \le poly(q)$.

### 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 $p$, $p \le poly(p)$. In other words, if a group satisfies property $p$ it also satisfies property $poly(p)$.

### 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 $p$, $poly(poly(p)) = poly(p)$. In other words, applying the poly operator twice has the same effect as applying it once.