Abstract

The distinguishing number is an important invariant used to measure the extent to which symmetries of graphs and permutation group actions can be broken by vertex labelings. In this paper, we investigate distinguishing labelings arising from permutation group actions with particular emphasis on Cartesian power constructions and wreath product actions. We establish structural bounds for distinguishing numbers in terms of orbit structure, stabilizers, and base size of permutation groups. Furthermore, we analyze the behavior of distinguishing numbers under Cartesian powers of sets and derive bounds for wreath product actions of the form \( G \wr S_m \) acting on \( X^m \). Several examples involving symmetric groups are presented to illustrate the theoretical results. These results contribute to a deeper understanding of symmetry breaking in permutation group actions and provide new insights into the interplay between distinguishing numbers and algebraic structures arising from wreath products.