## ABSTRACT

In this section we consider factorization of a periodic subset instead of a group. Plainly, a group distinct from {0} is always periodic, so we deal with an extension of factoring groups.

Let G be a finite abelian group and let A be a subset of G. We say that A is weakly periodic if there is an element g ∈ G \ {0} such that A and g + A mutually differ in at most one element. In other words, either A = g+A and so A is periodic or A contains one element that is not in g+A and conversely g + A contains one element that is not in A. The element g is called a weak period of A. Note that if g is a weak period of A, then so is −g. It is clear that a periodic subset is weakly periodic. Also, a cyclic subset is weakly periodic.