On Zero Sum Subsequences of Restricted Size $III$

Abstract

Let $p>2$ be a prime, and $G=C_{p^{e_1}}\oplus \dots \oplus C_{p^{e_k}}$ ($1\le e_1\le \cdots \le e_k$) 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 $p^{e_k}\ge 1+\sum _{i=1}^{k-1}(p^{e_i}–1)$, 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 $p^{e_k}$.