On Short Zero-sum Subsequences of Zero-sum Sequences

Abstract

Let $G$ be a finite abelian group of exponent $\exp(G)$. By $D(G)$ we denote the smallest integer $d\in \mathbb N$ such that every sequence over $G$ of length at least $d$ contains a nonempty zero-sum subsequence. By $\eta(G)$ we denote the smallest integer $d\in \mathbb N$ such that every sequence over $G$ of length at least $d$ contains a zero-sum subsequence $T$ with length $|T|\in [1,\exp(G)]$, such a sequence $T$ will be called a short zero-sum sequence. Let $C_0(G)$ denote the set consists of all integer $t\in [D(G)+1,\eta(G)-1]$ such that every zero-sum sequence of length exactly $t$ contains a short zero-sum subsequence. In this paper, we investigate the question whether $C_0(G)\neq \emptyset$ for all non-cyclic finite abelian groups $G$. Previous results showed that $C_0(G)\neq \emptyset$ for the groups $C_n^2$ ($n\geq 3$) and $C_3^3$. We show that more groups including the groups $C_m\oplus C_n$ with $3\leq m\mid n$, $C_{3^a5^b}^3$, $C_{3\times 2^a}^3$, $C_{3^a}^4$ and $C_{2^b}^r$ ($b\geq 2$) have this property. We also determine $C_0(G)$ completely  for some groups including the groups of rank two, and some special groups with large exponent.