# Word Maps in Finite Simple Groups

## TL;DR

This paper advances the understanding of word maps in finite simple groups by showing they can produce any endomorphism-closed subset, introduces a specific word for Mathieu group M11, and clarifies limitations of word map images.

## Contribution

It improves Lubotzky's result by allowing more flexible choices of words and provides a concrete example related to Mathieu group M11.

## Key findings

- Any endomorphism-closed subset of a finite simple group can be realized as a word map image.
- A specific word witnesses the chirality of Mathieu group M11.
- Not all endomorphism-closed subsets are images of word maps.

## Abstract

Elements of the free group define interesting maps, known as word maps, on groups. It was previously observed by Lubotzky that every subset of a finite simple group that is closed under endomorphisms occurs as the image of some word map. We improve upon this result by showing that the word in question can be chosen to be in any $v(\textbf{F}_n)$ provided that $v$ is not a law on the finite simple group in question. In addition, we provide an example of a word $w$ that witnesses the chirality of the Mathieu group $M_{11}$. The paper concludes by demonstrating that not every subset of a group closed under endomorphisms occurs as the image of a word map.