Distinguishing numbers for graphs and groups

A graph G is distinguished if its vertices are labelled by a map φ : V (G) −→ {1, 2, . . . , k} so that no non-trivial graph automorphism preserves φ. The distinguishing number of G is the minimum number k necessary for φ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of Γ on a set X. A labelling φ : X −→ {1, 2, . . . , k} is distinguishing if no element of Γ preserves φ except those which fix each element of X. The distinguishing number of the group action on X is the minimum k needed for φ to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of Sn on a set with distinguishing number n, answering an open question of Albertson and Collins.