Images of word maps in finite simple groups

TL;DR

This paper characterizes which subsets of finite simple groups can be realized as the image of a word map, showing they must contain the identity and be automorphism-invariant.

Contribution

It provides a complete characterization of subsets that are images of word maps in finite simple groups, answering a question posed by Kassabov, Nikolov, and Shalev.

Findings

A subset containing the identity and invariant under automorphisms is the image of some word map.

The characterization applies to all finite simple groups.

The result answers a previously open question.

Abstract

In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset $A$ of a finite simple group $G$ is the image of some word map $w : G\times G\to G $ if and only if (i) $A$ contains the identity and (ii) $A$ is invariant under $\Aut (G)$.