with 1 < n1 | … | nr be a finite abelian group, d*(G) = n1 +…+nr −r, and let d(G) denote the maximal length of a zerosum free sequence over G. Then d(G) ≥ d*(G), and the standing conjecture is that equality holds for G = Cnr. We show that equality does not hold for C2 ⊕ C2nr, where n ≥ 3 is odd and r ≥ 4. This gives new information on the structure of extremal zero-sum free sequences over C2nr.
Authors:Alfred Geroldinger, David J. Grynkiewicz, and Wolfgang A. Schmid
For a finite abelian group G and a positive integer d, let sdℕ(G) denote the smallest integer ℓ∊ℕ0 such that every sequence S over G of length |S|≧ℓ has a nonempty zero-sum subsequence T of length |T|≡0 mod d. We determine sdℕ(G) for all d≧1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erdős–Ginzburg–Ziv constant provided that, for the p-subgroups Gp of G, the Davenport constant D(Gp) is bounded above by 2exp (Gp)−1. This generalizes former results for groups of rank two.
Authors:Rui Chi, Shuyan Ding, Weidong Gao, Alfred Geroldinger, and Wolfgang A. Schmid
Summary For a finite abelian group G, we investigate the invariant s(G) (resp. the invariant s0(G)) which is defined as the smallest integer l ? N such that every sequence S in G of length |S| = l has a subsequence T with sum zero and length |T|= exp(G) (resp. length |T|=0 mod exp(G)).