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

TL;DR

This paper investigates the maximal cross numbers of minimal zero-sum and zero-sum free sequences in finite abelian groups, extending previous results and introducing new methods for specific group structures.

Contribution

It extends known results on cross numbers to new classes of finite abelian groups and develops a novel approach for verifying conjectured values in these groups.

Findings

Proved the conjecture for $ ext{k}(G)$ in certain group extensions.

Developed a new method for $ ext{K}(G)$ in specific abelian groups.

Provided structural insights into minimal zero-sum sequences.

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 $G \bigoplus C_{p^a} \bigoplus C_{p^b}$ 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.