On some weighted zero-sum constants

Abstract

Let G be a finite abelian group with exponent exp(G). Let A = {a ∈ ℤ : gcd(a,exp(G))=1}. The constant sA(G) is defined as the least positive integer t such that for any given sequence S of elements of G with length |S| ≥ ℓ it has a exp(G) length A-weighted zero-sum subsequence. In this article, we obtain the exact value of sA(G) for G = ℤpα ⊕ ℤp and an upper bound for the case G = ℤn ⊕ ℤp, where p is an odd prime, n is an odd integer and p | n. We also obtain the structural information on the extremal zero-sum free sequences.