The number of k-sums modulo k

Let a(1), ..., a(r) be a sequence of elements of Z(k), the integers module k. Calling the sum of Ic terms of the sequence a k-sum, how small can the s et of k-sums be? Our aim in this paper is to show that if 0 is not a k-sum then there are at least r - k + 1 k-sums. This result, which is best possib le, extends the Erdos-Ginzburg-Ziv theorem, which states that if r = 2k - 1 then 0 is a k-sum. We also show that the same result holds in any abelian group of order k, and make some related conjectures, (C) 1999 Academic Pres s.