Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group II

Abstract

Let G be an additive finite abelian group. Let F(G) denote the monoid of all sequences over G. For Ω⊂F(G), let dΩ(G) be the smallest integer t such that every sequence S over G of length |S|≥t has a subsequence in Ω. A sequence S over G is a weak-regular sequence if vg(S)≤ord(g) for every g∈G. Let N(G) be the smallest integer t such that every weak-regular sequence S over G of length |S|≥t has a nontrivial zero-sum subsequence T with vg(T)=vg(S)>0 for some g|S. The invariant N(G) was formulated in a recent paper [4] to study dΩ(G) and has its own interest, and N(G) has been determined for the cyclic groups of prime order there. In this paper, among other results, we shall determine the precise value of N(G) for cyclic groups of prime power order and for elementary abelian groups of rank two.