We introduce a group naturally acting on aperiodic necklaces of length n with two colors using a one-to-one correspondence between such necklaces and irreducible polynomials of degree n over the field F2 of two elements. We notice that this group is isomorphic to the quotient group of nondegenerate circulant matrices of size n over that field modulo a natural cyclic subgroup. Our groups turn out to be isomorphic to the sandpile groups for a special sequence of directed graphs. Bibliography: 15 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 421, 2014, pp. 81–93.
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Duzhin, S.V., Pasechnik, D.V. Groups Acting on Necklaces and Sandpile Groups. J Math Sci 200, 690–697 (2014). https://doi.org/10.1007/s10958-014-1960-6
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DOI: https://doi.org/10.1007/s10958-014-1960-6