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.

For two natural numbers 1 < p1 < p2, with α = log(p 1 ) log(p 2 ) irrational, we describe, in main Theorem 3.3 and in Note 3.5, the factorization of two adjacent numbers in the multiplicatively closed subset S = {p i i p j 2 | i, j ∈ N ∪ {0}} using primary and secondary convergents of α. This suggests general Question 1.2 for more than two generators which is still open.

No abstract

No abstract

No abstract