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On the Davenport constant and group algebras

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Smertnig D.

Colloquium Mathematicum

2010

tom 121

nr 2

179-193

For a finite abelian group G and a splitting field K of G, let 𝖽(G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g₁ · ... · g_{l}$ over G such that $(X^{g₁} - a₁) · ... · (X^{g_{l}} - a_{l}) ≠ 0 ∈ K[G]$ for all $a₁, ..., a_{l} ∈ K^{×}$. If 𝖣(G) denotes the Davenport constant of G, then there is the straightforward inequality 𝖣(G) - 1 ≤ 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 𝖣(G) - 1 < 𝖽(G,K). Thus we disprove the conjecture.

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1 Colloquium Mathematicum

1 Smertnig D.

1 2010

Wyniki wyszukiwania

On the Davenport constant and group algebras