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Báo cáo toán học: "The maximum distinguishing number of a group"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The maximum distinguishing number of a group. | The maximum distinguishing number of a group Melody Chan Princeton University Princeton New Jersey USA melody.chan@aya.yale.edu Submitted Sep 9 2004 Accepted Feb 10 2006 Published Aug 7 2006 Mathematics Subject Classification 05E15 20B25 20D60 Abstract 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 show that if G is nilpotent of class c or supersolvable of length c then G always acts with distinguishing number at most c 1. We obtain that all metacyclic groups act with distinguishing number at most 3 these include all groups of squarefree order. We also prove that the distinguishing number of the action of the general linear group GLn K over a field K on the vector space Kn is 2 if K has at least n 1 elements. 1 Introduction An action of a group G on a set X is said to be faithful if only the identity element of G fixes every element of X. Let G be a group acting faithfully on X. For r 2 N an r-coloring of X is a function c X 1 . r . A permutation of X preserves the coloring c if c x c x for all x 2 X .A coloring is said to be distinguishing if the only element in G that induces a color-preserving permutation of X is the identity element. The distinguishing number of the action of G on X denoted Dg X is the smallest r admitting a distinguishing r-coloring of X with respect to the action of G. If there does not exist a distinguishing r-coloring of X for any finite r we say that Dg X 1. The distinguishing number was first defined by Albertson and Collins in 2 as a property of graphs. More specifically the distinguishing number of a graph M denoted D M is the smallest number of colors admitting a coloring of the vertices such that the only color-preserving automorphism of M is the identity thus D M DAut M V M . Note that distinguishing .