The distinguishing number of the direct product and wreath product action |
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Authors: | Melody Chan |
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Affiliation: | (1) University of Cambridge, Cambridge, England |
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Abstract: | Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted D G(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product G ≀Y H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups S m × S n on [m] × [n]. |
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Keywords: | Symmetry group Symmetry breaking Distinguishing number Wreath product Direct product |
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