# On the Symmetry of Images of Word Maps in Groups

## TL;DR

This paper investigates the symmetry of images of word maps in groups, characterizing conditions for their closure under inversion, and identifying small groups with non-symmetric word map images, especially in nilpotent groups.

## Contribution

It provides new criteria for groups to have symmetric word map images and classifies small groups with non-symmetric images, extending understanding of word map behavior.

## Key findings

- Only two groups under 108 elements have non-symmetric word map images.
- Certain conditions guarantee all word map images are closed under inversion.
- Analysis of nilpotent groups reveals similar symmetry properties.

## Abstract

Word maps in a group, an analogue of polynomials in groups, are defined by substitution of formal words. Lubotzky gave a characterization of the images of word maps in finite simple groups, and a consequence of his characterization is the existence of a group G such that the image of some word map on G is not closed under inversion. We explore sufficient conditions on a group that ensure that the image of all word maps on G are closed under inversion. We then show that there are only two groups with order less than 108 with the property that there is a word map with image not closed under inversion. We also study this behavior in nilpotent groups.