Distinguishing labeling of group actions

Suppose Γ is a group acting on a set X. An r-labeling f:X→{1,2,…,r} of X is distinguishing (with respect to Γ) if the only label preserving permutation of X in Γ is the identity. The distinguishing number, D_Γ(X), of the action of Γ on X is the minimum r for which there is an r-labeling which is distinguishing. This paper investigates the relation between the cardinality of a set X and the distinguishing numbers of group actions on X. For a positive integer n, let D(n) be the set of distinguishing numbers of transitive group actions on a set X of cardinality n, i.e., D(n)={D_Γ(X):|X|=n and Γ acts transitively on X}. We prove that . Then we consider the problem of an arbitrary fixed group Γ acting on a large set. We prove that if for any action of Γ on a set Y, for each proper normal subgroup H of Γ, D_H(Y)≤2, then there is an integer n such that for any set X with |X|≥n, for any action of Γ on X with no fixed points, D_Γ(X)≤2.