  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.
