Word Maps in Finite Simple Groups

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.