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 $H=C_{p^{c_1}}\oplus \dots C_{p^{c_l}}$ be a $p$-group. Suppose that $1+\sum _{i=1}^l(p^{c_i}-1)=p^k$ for some positive integer $k$. Then,$4p^k–3\le s(C_{p^k}\oplus H)\le 4p^k–2.$