Zero-sum invariants on finite abelian groups with large exponent

Let $G$ be a finite abelian group of exponent $\exp(G)$. By $D(G)$ we denote the smallest integer $d\in \mathbb N$ such that every sequence over $G$ of length at least $d$ contains a nonempty zero-sum subsequence. By $\eta(G)$ we denote the smallest integer $d\in \mathbb N$ such that every sequence over $G$ of length at least $d$ contains a zero-sum subsequence $T$ with length $|T|\in [1,\exp(G)]$, such a sequence $T$ will be called a short zero-sum sequence. Let $C_0(G)$ denote the set consists of all integer $t\in [D(G)+1,\eta(G)-1]$ such that every zero-sum sequence of length exactly $t$ contains a short zero-sum subsequence. In this paper, we investigate the question whether $C_0(G)\neq \emptyset$ for all non-cyclic finite abelian groups $G$. Previous results showed that $C_0(G)\neq \emptyset$ for the groups $C_n^2$ ($n\geq 3$) and $C_3^3$. We show that more groups including the groups $C_m\oplus C_n$ with $3\leq m\mid n$, $C_{3^a5^b}^3$, $C_{3\times 2^a}^3$, $C_{3^a}^4$ and $C_{2^b}^r$ ($b\geq 2$) have this property. We also determine $C_0(G)$ completely for some groups including the groups of rank two, and some special groups with large exponent.

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On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups

Colloquium Mathematicum ◽

10.4064/cm8033-12-2019 ◽

2021 ◽

Vol 163 (2) ◽

pp. 317-332

Author(s):

Jiangtao Peng ◽

Yuanlin Li ◽

Chao Liu ◽

Meiling Huang

Keyword(s):

Inverse Problems ◽

Abelian Groups ◽

Finite Abelian Groups ◽

Zero Sum

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Two zero-sum invariants on finite abelian groups

European Journal of Combinatorics ◽

10.1016/j.ejc.2013.05.018 ◽

2013 ◽

Vol 34 (8) ◽

pp. 1331-1337 ◽

Cited By ~ 12

Author(s):

Yushuang Fan ◽

Weidong Gao ◽

Linlin Wang ◽

Qinghai Zhong

Keyword(s):

Abelian Groups ◽

Finite Abelian Groups ◽

Zero Sum

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On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups Ⅱ

AIMS Mathematics ◽

10.3934/math.2021101 ◽

2021 ◽

Vol 6 (2) ◽

pp. 1706-1714

Author(s):

Rui Wang ◽

◽

Jiangtao Peng

Keyword(s):

Inverse Problems ◽

Abelian Groups ◽

Finite Abelian Groups ◽

Zero Sum

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Unextendible Sequences in Finite Abelian Groups

The Electronic Journal of Combinatorics ◽

10.37236/899 ◽

2008 ◽

Vol 15 (1) ◽

Author(s):

Jujuan Zhuang

Keyword(s):

Abelian Group ◽

Abelian Groups ◽

Finite Abelian Group ◽

Minimal Length ◽

Finite Abelian Groups ◽

Zero Sum

Let $G=C_{n_1}\oplus \ldots \oplus C_{n_r}$ be a finite abelian group with $r=1$ or $1 < n_1|\ldots|n_r$, and let $S=(a_1,\ldots,a_t)$ be a sequence of elements in $G$. We say $S$ is an unextendible sequence if $S$ is a zero-sum free sequence and for any element $g\in G$, the sequence $Sg$ is not zero-sum free any longer. Let $L(G)=\lceil \log_2{n_1}\rceil+\ldots+\lceil \log_2{n_r}\rceil$ and $d^*(G)=\sum_{i=1}^r(n_i-1)$, in this paper we prove, among other results, that the minimal length of an unextendible sequence in $G$ is not bigger than $L(G)$, and for any integer $k$, where $L(G)\leq k \leq d^*(G)$, there exists at least one unextendible sequence of length $k$.

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