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 | D ( n ) | = O ( n ) . 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 .