Taiwanese Journal of Mathematics

MINIMAL ZERO-SUM SEQUENCES IN FINITE CYCLIC GROUPS

Jujuan Zhuang and Pingzhi Yuan

Full-text: Open access

Abstract

Let $C_n$ be the cyclic group of order $n$, $n\geq 20$, and let $S=\prod_{i=1}^k g_i$ be a minimal zero-sum sequence of elements in $C_n$. We say that $S$ is insplitable if for any $g_i\in S$ and any two elements $x,y\in C_n$ satisfying $x+y=g_i$, $Sg_i^{-1}xy$ is not a minimal zero-sum sequence any more. We define $\mbox{Index}(S)=\min_{(m,n)=1} \{\sum_{i=1}^k|mg_i|\}$, where $|x|$ denotes the least positive inverse image under homomorphism from the additive group of integers $\mathbb{Z}$ onto $C_n$. In this paper we prove that for an insplitable minimal zero-sum sequence $S$, if $\mbox{Index}(S)=2n$, then $|S|\leq \lfloor\frac{n}{2}\rfloor+1$.

Article information

Source
Taiwanese J. Math., Volume 13, Number 3 (2009), 1007-1015.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405455

Digital Object Identifier
doi:10.11650/twjm/1500405455

Mathematical Reviews number (MathSciNet)
MR2526354

Zentralblatt MATH identifier
1189.11009

Subjects
Primary: 11B50: Sequences (mod $m$) 11p21

Keywords
minimal zero-sum sequence finite cyclic group

Citation

Zhuang, Jujuan; Yuan, Pingzhi. MINIMAL ZERO-SUM SEQUENCES IN FINITE CYCLIC GROUPS. Taiwanese J. Math. 13 (2009), no. 3, 1007--1015. doi:10.11650/twjm/1500405455. https://projecteuclid.org/euclid.twjm/1500405455


Export citation

References

  • Scott T. Chapman, Michael Freeze, and William W.Smith, Minimal zero-sequences and the strong Davenport constant, Discrete Math., 203 (1999), 271-277.
  • Weidong Gao, Zero-sums in finite cyclic groups, Integers: Electronic J. Combinatorial Number Theory, 2000.
  • A. Geroldinger and F. Halter-Koch,Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Chapman & Hall/CRC, Boca Raton, 2006.
  • Svetoslav Savchev, Fang Chen, Long zero-free sequences in finite cyclic groups, Discrete Mathematics (2007), doi: 10.1016/j.disc.2007.01.012.
  • Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, J. Combinatorial Theory $($Series A$)$, 114 (2007), 1545-1551.
  • V. Ponomarenko, Minimal zero sequences of finite cyclic groups, INTEGERS: Electronic J. Combinatorial Number Theory, 2004, p. 4.