On the Davenport constant and group algebras

Daniel Smertnig

Daniel Smertnig

Colloquium Mathematicae, Tome 120 (2010), p. 179-193 / Harvested from The Polish Digital Mathematics Library

Résumé

For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (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 (G) - 1 < (G,K). Thus we disprove the conjecture.

Publié le : 2010-01-01

Zbl 1206.11128

EUDML-ID : urn:eudml:doc:286476

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     title = {On the Davenport constant and group algebras},
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Daniel Smertnig. On the Davenport constant and group algebras. Colloquium Mathematicae, Tome 120 (2010) pp. 179-193. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-2-2/