The distinguishing number of groups based on the distinguishing number of subgroups

2016/12/31 by Alikhani, Saeid, Soltani, Samaneh
#05C15 #05E15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1701.00141

Abstract

Let Γ be a group acting on a set X. The distinguishing number for this action of Γ on X, denoted by D_Γ(X), is the smallest natural number k such that the elements of X can be labeled with k labels so that any label-preserving element of Γ fixes all x ∈ X. In particular, if the action is faithful, then the only element of Γ preserving labels is the identity. In this paper, we obtain an upper bound on the distinguishing number of a set knowing the distinguishing number of a set under the action of a subgroup. By the concept of motion, we obtain an upper bound for the distinguishing number of a group. Motivated by a problem (Chan 2006), we characterize D_Γ,H(X) which is the smallest number of labels admitting a labeling of X such that the only elements of Γ that induce label-preserving permutations lie in H. Finally, we state two algorithms for obtaining an upper and a lower bound for D_{Γ, H}(X).

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