On the Davenport constant and on the structure of extremal zero-sum free sequences

Abstract

Let $G=C_{n_1}\oplus ...\oplus C_{n_r}$ with $1 < n_1 \t ... \t n_r$ be a finite abelian group, ${\mathsf{d}}^{\ast }(G)=n_1+...+n_r-r$, and let $\mathsf{d}(G)$ denote the maximal length of a zero-sum free sequence over $G$. Then $\mathsf{d}(G)\ge {\mathsf{d}}^{\ast }(G)$, and the standing conjecture is that equality holds for $G=C_n^r$. We show that equality does not hold for $C_2\oplus C_{2n}^r$, where $n\ge 3$ is odd and $r\ge 4$. This gives new information on the structure of extremal zero-sum free sequences over $C_{2n}^r$.

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