The distinguishing number of the direct product and wreath product action
نویسنده
چکیده
Let G be a group acting faithfully on a set X . The distinguishing number of the action of G on X , denoted DG(X ), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a colorpreserving 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 Sm × Sn on [m] × [n].
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تاریخ انتشار 2006