  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 $O\times O$ under the diagonal action
of the automorphism group.
