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

Abstract

The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by a trivial automorphism. 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 ϵ 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. Also we study 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.