On Short Zero-Sum Subsequences

Abstract

Let \(G\) be a finite abelian group of exponent \(m\). By \(s(G)\) we denote the smallest integer \(c\) such that every sequence of \(t\) elements in \(G\) contains a zero-sum subsequence of length \(m\). Among other results, we prove that, let \(p\) be a prime, and let \(H = C_{p^{c_1}} \oplus \ldots C_{p^{c_l}}\) be a \(p\)-group. Suppose that \(1+\sum_{i=1}^{l}(p^{c_i}-1)=p^k\) for some positive integer \(k\). Then,\(4p^k – 3 \leq s(C_{p^k} \oplus H) \leq 4p^k – 2.\)