On the Davenport constant and group algebras

Daniel Smertnig¹

¹ Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria

Colloquium Mathematicum, Tome 121 (2010) no. 2, pp. 179-193

Résumé

For a finite abelian group $G$ and a splitting field $K$ of $G$, let $\mathsf d (G, K)$ denote the largest integer $l \in \mathbb N$ for which there is a sequence $S = g_1 \cdot \ldots \cdot g_l$ over $G$ such that $(X^{g_1} - a_1) \cdot \ldots \cdot (X^{g_l} - a_l) \ne 0 \in K[G]$ for all $a_1, \ldots, a_l \in K^{\times}$. If $\mathsf D (G)$ denotes the Davenport constant of $G$, then there is the straightforward inequality $\mathsf D (G)-1 \le \mathsf d (G, K)$. Equality holds for a variety of groups, and a 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 $\mathsf D (G) -1 \mathsf d (G, K)$. Thus we disprove the conjecture.

Détail

DOI : 10.4064/cm121-2-2

Keywords: finite abelian group splitting field mathsf denote largest integer mathbb which there sequence cdot ldots cdot cdot ldots cdot ldots times mathsf denotes davenport constant there straightforward inequality mathsf mathsf equality holds variety groups conjecture gao nbsp states equality holds groups offer further groups which equality holds first examples groups which mathsf mathsf disprove conjecture