scholarly journals On Zero-sum Free Subsets of Length 7

10.37236/376 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Pingzhi Yuan ◽  
Xiangneng Zeng
Keyword(s):  
Abelian Group ◽  
Nonempty Subset ◽  
Zero Sum

Let $G$ be a finite additively written abelian group, and let $X$ be a subset of 7 elements in $G$. We show that if $X$ contains no nonempty subset with sum zero, then the number of the elements which can be expressed as the sum over a nonempty subsequence of $X$ is at least $ 24$.

scholarly journals Subsums of a Zero-sum Free Subset of an Abelian Group

10.37236/840 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Weidong Gao ◽  
Yuanlin Li ◽  
Jiangtao Peng ◽  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Cyclic Group ◽  
Open Problem ◽  
Nonempty Subset ◽  
Finite Abelian Group ◽  
Positive Answer ◽  
Free Subset ◽  
Zero Sum ◽  
G 28

Let $G$ be an additive finite abelian group and $S \subset G$ a subset. Let f$(S)$ denote the number of nonzero group elements which can be expressed as a sum of a nonempty subset of $S$. It is proved that if $|S|=6$ and there are no subsets of $S$ with sum zero, then f$(S)\geq 19$. Obviously, this lower bound is best possible, and thus this result gives a positive answer to an open problem proposed by R.B. Eggleton and P. Erdős in 1972. As a consequence, we prove that any zero-sum free sequence $S$ over a cyclic group $G$ of length $|S| \ge {6|G|+28\over19}$ contains some element with multiplicity at least ${6|S|-|G|+1\over17}$.

On subset sums of zero-sum free sets of abelian groups

2019 ◽  
Vol 15 (03) ◽  
pp. 645-654 ◽  
Author(s):  
Jiangtao Peng ◽  
Wanzhen Hui ◽  
Yuanlin Li ◽  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Nonempty Subset ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Subset Sums ◽  
Zero Sum ◽  
Free Sets

Let [Formula: see text] be a finite abelian group and [Formula: see text] be a subset of [Formula: see text]. Let [Formula: see text] denote the set of group elements which can be expressed as a sum of a nonempty subset of [Formula: see text]. We say that [Formula: see text] is zero-sum free if [Formula: see text]. In this paper, we show that if [Formula: see text] is zero-sum free with [Formula: see text], then [Formula: see text]. Furthermore, if [Formula: see text] is zero-sum free and [Formula: see text], with the assumption that the subgroup generated by [Formula: see text] is noncyclic and the order [Formula: see text], we are able to show that [Formula: see text].

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

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

scholarly journals On Subsequence Sums of a Zero-sum Free Sequence

10.37236/970 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Open Question

Let $G$ be a finite abelian group with exponent $m$, and let $S$ be a sequence of elements in $G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we show that, if $|S|=m$ and $S$ contains no nonempty subsequence with zero sum, then $f(S)\geq 2m-1$. This answers an open question formulated by Gao and Leader. They proved the same result with the restriction $(m,6)=1$.

On the Maximal Cross Number of Unique Factorization Zero-Sum Sequences over a Finite Abelian Group

Integers ◽  
2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Weidong Gao ◽  
Linlin Wang
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Unique Factorization ◽  
The Cross ◽  
Cross Number ◽  
Zero Sum

Abstract.Letdenote the cross number ofWe determine

A unifying look at zero-sum invariants

2018 ◽  
Vol 14 (03) ◽  
pp. 705-711
Author(s):  
Weidong Gao ◽  
Yuanlin Li ◽  
Jiangtao Peng ◽  
Guoqing Wang
Keyword(s):  
Abelian Group ◽  
Open Problems ◽  
First Results ◽  
Zero Sum

The purpose of the paper is to provide a unified way to formulate zero-sum invariants. Let [Formula: see text] be a finite additive abelian group. Let [Formula: see text] denote the monoid of all zero-sum sequences over [Formula: see text]. For [Formula: see text], let [Formula: see text] be the smallest integer [Formula: see text] such that every sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] has a subsequence in [Formula: see text]. We provide some first results and open problems on [Formula: see text].

On zero-sum subsequences of prescribed length

2017 ◽  
Vol 14 (01) ◽  
pp. 167-191 ◽  
Author(s):  
Dongchun Han ◽  
Hanbin Zhang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Zero Sum

Let [Formula: see text] be an additive finite abelian group with exponent [Formula: see text]. For any positive integer [Formula: see text], let [Formula: see text] be the smallest positive integer [Formula: see text] such that every sequence [Formula: see text] in [Formula: see text] of length at least [Formula: see text] has a zero-sum subsequence of length [Formula: see text]. Let [Formula: see text] be the Davenport constant of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite abelian [Formula: see text]-group with [Formula: see text] then [Formula: see text] for every [Formula: see text], which confirms a conjecture by Gao et al. recently, where [Formula: see text] is a prime.

scholarly journals Zero-sum subsets of decomposable sets in Abelian groups

2020 ◽  
Vol 30 (1) ◽  
pp. 15-25
Author(s):  
T. Banakh ◽  
◽  
A. Ravsky ◽  
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Open Problem ◽  
Abelian Groups ◽  
The Other ◽  
Other Hand ◽  
Decomposable Sets ◽  
Empty Subset ◽  
Zero Sum

A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.

scholarly journals On the generalized Davenport constant and the Noether number

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
Kálmán Cziszter ◽  
Mátyás Domokos
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Polynomial Invariants ◽  
Arbitrary Finite Group ◽  
Zero Sum ◽  
Zero Vector ◽  
Degree Bound

AbstractKnown results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.

On some weighted zero-sum constants

2017 ◽  
Vol 13 (02) ◽  
pp. 301-308 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structural Information ◽  
Finite Abelian Group ◽  
Text Length ◽  
Zero Sum

Let [Formula: see text] be a finite abelian group with exponent exp[Formula: see text]. Let [Formula: see text]. The constant [Formula: see text] is defined as the least positive integer [Formula: see text] such that for any given sequence [Formula: see text] of elements of [Formula: see text] with length [Formula: see text] it has a [Formula: see text] length [Formula: see text]-weighted zero-sum subsequence. In this article, we obtain the exact value of [Formula: see text] for [Formula: see text] and an upper bound for the case [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] is an odd integer and [Formula: see text]. We also obtain the structural information on the extremal zero-sum free sequences.

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