On the Davenport constant and group algebras

TL;DR

This paper investigates the relationship between the Davenport constant and a new invariant related to group algebras, providing counterexamples that disprove a standing conjecture and expanding understanding of these algebraic properties.

Contribution

It introduces the invariant d(G, K), compares it with the Davenport constant, and provides the first known examples where the conjectured equality does not hold.

Findings

Established cases where equality holds between D(G)-1 and d(G, K)

Provided the first examples where D(G)-1 < d(G, K), disproving the conjecture

Extended the class of groups for which the equality is known to hold

Abstract

For a finite abelian group $G$ and a splitting field $K$ of $G$, let $d(G, K)$ denote the largest integer $l \in \N$ for which there is a sequence $S = g_1 \cdot ... \cdot g_l$ over $G$ such that $(X^{g_1} - a_1) \cdot ... \cdot (X^{g_l} - a_l) \ne 0 \in K[G]$ for all $a_1, ..., a_l \in K^{\times}$. If $D(G)$ denotes the Davenport constant of $G$, then there is the straightforward inequality $D(G)-1 \le d (G, K)$. Equality holds for a variety of groups, and a standing conjecture of W. Gao et.al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups $G$ for which $D(G) -1 < d(G, K)$ holds. Thus we disprove the conjecture.