Geodesic
Note on the Davenport constant for finite abelian groups with rank three
Maciej Szymon Zakarczemny1 ; Maciej Szymon Zakarczemny
1Department of Applied Mathematics, Faculty of Computer Science and Telecommunications, Cracow University of Technology, Cracow, Poland,
Acta mathematica Universitatis Comenianae, Tome 90 (2021) no. 1, pp. 1-6 
Maciej Szymon Zakarczemny; Maciej Szymon Zakarczemny. Note on the Davenport constant for finite abelian groups with rank three. Acta mathematica Universitatis Comenianae, Tome 90 (2021) no. 1, pp. 1-6. http://geodesic.mathdoc.fr/item/AMUC_2021_90_1_a0/
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Voir la notice de l'article provenant de la source Comenius University

Let G be a finite abelian group and D(G) denote the Davenport constant of G. We derive new upper bound for the Davenport constant for all finite abelian groups of rank three. Our main result is that: D(Cn1 ⊕ Cn2 ⊕ Cn3 ) ≤ (n1 − 1) + (n2 − 1) + (n3 − 1) + 1 + (a3 − 3)(n1 − 1), where 1 < n1|n2|n3 ∈ N and a3 ≤ 20369 is a constant. Therefore D(Cn1 ⊕ Cn2 ⊕ Cn3 ) grows linearly with the variables n1, n2, n3. The new result is the given upper bound for a3. Finally, we give an application of the Davenport constant to smooth numbers.