Permutation representations of the orbits of the automorphism group of a finite module over discrete valuation ring

Abstract

Abstract. Consider a discrete valuation ring R whose residue field is finite of cardinality at least 3. For a finite torsion module, we consider transitive subsets O under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space C[O] is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of O×O under the diagonal action of the automorphism group.