Multifold factorizations of cyclic groups into subsets

Let (G, +) be an abelian group. A finite multiset A over G is said to give a A-fold factorization of G if there exists a multiset B over G such that each element of G occurs A times in the multiset A+B := {a+b : a is an element of A, b is an element of B}. In this article, restricting G to a cyclic group, we will provide sufficient conditions on a given multiset A under which the exact value or an upper bound of the minimum multiplicity lambda of a factorization of G can be given by introducing a concept of 'lcm-closure'. Furthermore, a couple of properties on a given factor A will be shown when A has a prime or prime power order (cardinality). A relation to multifold factorizations of the set of integers will be also glanced at a general perspective. (C) 2018 Elsevier Inc. All rights reserved.