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arXiv · 1701.05947

On the Symmetry of Images of Word Maps in Groups

William Cocke·Meng-Che Ho
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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.

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William Cocke, Meng-Che Ho. 2017-01-20. On the Symmetry of Images of Word Maps in Groups. https://arxiv.org/abs/1701.05947

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