Multifold factorizations of cyclic groups into subsets

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Abstract

Let (G,+) be an abelian group. A finite multiset A over G is said to give a λ-fold factorization of G if there exists a multiset B over G such that each element of G occurs λ times in the multiset A+B:={a+b:aA,bB}. 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 λ 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.

MSC

05B45
11B75

Keywords

Factorizations of cyclic groups
Multifold factorizations
Cyclotomic polynomials
Complement factor problem
Periodic factor

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