On Zero Sum Subsequences of Restricted Size \(III\)

Abstract

Let \(p > 2\) be a prime, and \(G = C_{p^{e_1}} \oplus \ldots \oplus C_{p^{e_k}}\) (\(1 \leq e_1 \leq \cdots \leq e_k\)) a finite abelian \(p\)-group. We prove that \(1 + 2\sum_{i=1}^{k}(p^{e_i} – 1)\) is the smallest integer \(t\) such that every sequence of \(t\) elements in \(G\) contains a zero-sum subsequence of odd length. As a consequence, we derive that if \(p^{e_k} \geq 1 + \sum_{i=1}^{k-1} (p^{e_i} – 1)\), then every sequence of \(4p^{e_k} – 3 + 2\sum_{i=1}{k-1} (p^{e_i} – 1)\) elements in \(G\) contains a zero-sum subsequence of length \(p^{e_k}\).