Note on the Davenport's constant for finite abelian groups with rank three

Abstract

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 groups of rank three. Our main result is that:where 1 < n 1 |n 2 |n 3 ∈ N and a 3 ≤ 20369 is a constant. Therefore D(C n 1 ⊕ C n 2 ⊕ C n 3 ) grows linearly with the variables n 1 , n 2 , n 3 . The new result is the given upper bound for a 3 . Finally, we give an application of the Davenport constant to smooth numbers.