The Cross Number of Minimal Zero-sum Sequences in Finite Abelian Groups

Abstract: 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 $$G \bigoplus C_{p^a} \bigoplus C_{p<sup>b}$$ 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 $H_p \bigoplus C_{q<sup>m}$ (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.