On the symmetry of images of word maps in groups

ABSTRACT

Word maps on a group 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 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.

KEYWORDS:

• Nilpotent groups
• word maps

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20F10
• 20F28

Acknowledgments

The authors thank Dan Segal for his correspondence throughout the project. The authors would also like to thank the anonymous referee for his helpful comments and suggestions. This material is based upon work suppored by the National Science Foundation Graduate Researach Fellowship Program under Grant No. DGE-1256529.