Note on a Zero-Sum Problem

Let G be an additively written, finite abelian group, and exp(G ) its exponent. Let S=(a1 , ..., ak) be a sequence of elements in G; we say that S is a zero-sum sequence if i=1 ai=0. Let s(G ) be the samllest integer t such that every sequence of t elements in G contains a zero-sum subsequence of length exp(G ). This constant has been studied by serveral authors during last 20 years [1 9, 11]. Let Cn be the cyclic group of order n, and C n the direct product of k copies of Cn . In [3], Erdo s et al. proved that S(C n)=2n&1. A geometrical interpretation of s(C n) was given by Harborth in [8]. In 1980, Kemntiz suggested the following