Remarks on a Zero-Sum Theorem

Abstract

Recently the following theorem in combinatorial group theory has been proved: LetGbe a finite abelian group and letAbe a sequence of members ofGsuch that |A|⩾|G|+D(G)−1, whereD(G) is the Davenport constant ofG. ThenAcontains a subsequenceBsuch that |B|=|G| and ∑b∈Bb=0. We shall present a generalization of this theorem which contains information on the extremal cases and in particular allows us to deduce a short proof of the extremal cases in the Erdős–Ginzburg–Ziv theorem. We also present, using the above-mentioned theorem, a proof that ifGhas rankkthen |A|⩾|G|(1+(k+1)/2k)−1 suffices to ensure a zero-sum subsequence on |G| terms.