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On Long Minimal Zero Sequences in Finite Abelian Groups

Periodica Mathematica Hungarica
Volume/Issue:
Volume 38: Issue 3
DOI:
https://doi.org/10.1023/a:1004854508165
Authors:
Weidong Gao
and
Alfred Geroldinger

Without Abstract

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On the order of elements in long minimal zero-sum sequences

Periodica Mathematica Hungarica
Volume/Issue:
Volume 44: Issue 1
DOI:
https://doi.org/10.1023/a:1014923902367
Authors:
Weidong Gao
and
Alfred Geroldinger
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On the Davenport constant and on the structure of extremal zero-sum free sequences

Periodica Mathematica Hungarica
Volume/Issue:
Volume 64: Issue 2
DOI:
https://doi.org/10.1007/s10998-012-3378-6
Authors:
Alfred Geroldinger
,
Manfred Liebmann
, and
Andreas Philipp

Abstract  

Let
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$G = C_{n_1 } \oplus \cdots \oplus C_{n_r }$$ \end{document}
with 1 < n 1 | … | n r be a finite abelian group, d*(G) = n 1 +…+n r −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 = C n r . We show that equality does not hold for C 2 ⊕ C 2n r , where n ≥ 3 is odd and r ≥ 4. This gives new information on the structure of extremal zero-sum free sequences over C 2n r .
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Zero-sum problems with congruence conditions

Acta Mathematica Hungarica
Volume/Issue:
Volume 131: Issue 4
DOI:
https://doi.org/10.1007/s10474-011-0073-7
Authors:
Alfred Geroldinger
,
David J. Grynkiewicz
, and
Wolfgang A. Schmid

Abstract

For a finite abelian group G and a positive integer d, let s dℕ(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 s dℕ(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 G p of G, the Davenport constant D(G p ) is bounded above by 2exp  (G p )−1. This generalizes former results for groups of rank two.

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On zero-sum subsequences of restricted size. IV

Acta Mathematica Hungarica
Volume/Issue:
Volume 107: Issue 4
DOI:
https://doi.org/10.1007/s10474-005-0201-3
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)).

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