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A word w in a free group is achiral if for every group G, Gw=Gw−1, where Gw is the image of the word map w on G. We give few classes of examples of achiral words. We prove that all words in F2 of length at most 7 are achiral. We provide a criterion to decide whether a word in F2 is chiral or achiral. As a result, we give shortest chiral word in F2 which is of length 8 and we answer Cocke and Ho's query regarding whether Engel words are achiral.
Singh et al. (Tue,) studied this question.
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