Inverse zero-sum problems and algebraic invariants

TL;DR

This paper investigates the maximum cross number of long zero-sumfree sequences in finite Abelian groups, proposing a conjecture and proving it for specific group classes, with implications for algebraic invariants and sequence properties.

Contribution

It formulates a general conjecture on inverse zero-sum problems and proves it for cyclic, p-groups, and rank-two groups, also improving related bounds.

Findings

Conjecture holds for cyclic groups, p-groups, and rank-two groups.

Improves bounds on the minimal number of elements with maximal order.

Provides methods to analyze algebraic invariants of zero-sumfree sequences.

Abstract

In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of rank two.