Davenport’s constant for groups with large exponent

Abstract

Let G be a finite abelian group. We show that its Davenport constant D(G )s atisf iesD(G) ≤ exp(G )+ |G| exp(G) − 1, provided that exp(G) ≥ |G| ,a ndD(G) ≤ 2 |G |− 1, if exp(G) < |G|. This proves a conjecture by Balasubramanian and the first named author. |G| exp(G) , 7), and conjectured that one may replace the constant 7 by |G|. Here we prove this conjecture. It turns out that the hypothesis that k be integral creates some technical difficulties, therefore we prove the following, slightly sharper result. Theorem 1.1. For an abelian group G with exp(G) ≥ |G| we have D(G) ≤ exp(G )+ |G| exp(G) − 1 ,w hile forexp(G) < |G| we have D(G) ≤ 2 |G |− 1. We notice that the first upper bound is actually reached for groups of rank 2 where D(G )=e xp(G )+ |G| exp(G) − 1. An application of our bound to random groups and (Z ∗, ·) will be the topic of a forthcoming paper. Let s≤n(G) be the least integer k, such that every sequence of length k contains a subsequence of length ≤ n adding up to 0 and let s=n(G) be the least integer k such that any sequence of length k in G contains a zero-sum of sequence of length exactly equal to n. In the special case where n =e xp(G )w e use the more standard