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We study the maximal cross number $mathsf{K}(G)$ of a minimal zero-sum sequence and the maximal cross number $mathsf{k}(G)$ of a zero-sum free sequence over a finite abelian group $G$, defined by Krause and Zahlten. In the first part of this paper, we extend a previous result by X. He to prove that the value of $mathsf{k}(G)$ conjectured by Krause and Zahlten hold for $G bigoplus C_{p^a} bigoplus C_{p^b}$ when it holds for $G$, provided that $p$ and the exponent of $G$ are related in a specific sense. In the second part, we describe a new method for proving that the conjectured value of $mathsf{K}(G)$ hold for abelian groups of the form $H_p bigoplus C_{q^m}$ (where $H_p$ is any finite abelian $p$-group) and $C_p bigoplus C_q bigoplus C_r$ for any distinct primes $p,q,r$. We also give a structural result on the minimal zero-sum sequences that achieve this value.
Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1
Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/or
The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $Omega subset B(G$), let $d_{O
Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is uns
Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)cdotldotscdot(x_lg)$ where $gin G$ and $x_1, ldots, x_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(x_1