Short Zero-Sum Sequences Over Abelian $p$-Groups of Large Exponent

TL;DR

This paper determines the exact value of a zero-sum sequence invariant for certain abelian p-groups, confirming a conjecture and advancing understanding of zero-sum problems in finite abelian groups.

Contribution

It provides the precise value of η(G) for p-groups with Davenport constant at most 2n-1, confirming a conjecture by Schmid and Zhuang.

Findings

Calculated η(G) for specific p-groups

Confirmed a conjecture in zero-sum theory

Enhanced understanding of zero-sum sequences in abelian groups

Abstract

Let $G$ be a finite abelian group with exponent $n$. Let $\eta(G)$ denote the smallest integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a zero-sum subsequence of length at most $n$. We determine the precise value of $\eta(G)$ when $G$ is a $p$-group whose Davenport constant is at most $2n-1$. This confirms one of the equalities in a conjecture by Schmid and Zhuang from 2010.