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Inverse zero-sum problem of finite abelian groups of rank $2$

Volume 175 / 2024

Wanzhen Hui, Meiling Huang, Yuanlin Li Colloquium Mathematicum 175 (2024), 77-95 MSC: Primary 11B75; Secondary 11P70 DOI: 10.4064/cm9040-12-2023 Published online: 5 February 2024

Abstract

Let $G$ be a finite abelian group and $S$ be a sequence over $G$. Let $\Sigma _k(S)$ denote the set of group elements which can be expressed as a sum of a subsequence of $S$ with length $k$. We study $\Sigma _{n^2m}(S)$ of a sequence $S$ over $C_n\oplus C_{nm}$, where $n,m$ are positive integers and $|S|=n^2m+r$ with $r\in \{nm+n-4,nm+n-3\}$. We show that either $0\in \Sigma _{n^2m}(S)$ or $|\Sigma _{n^2m}(S)|\geq (r-nm+3)nm-1$. Furthermore, we determine the structure of $S$ if $0\notin \Sigma _{n^2m}(S)$ and $|\Sigma _{n^2m}(S)|= (r-nm+3)nm-1$.

Authors

  • Wanzhen HuiSchool of Mathematical Sciences
    Sichuan Normal University
    Chengdu, 610066, P.R. China
    e-mail
  • Meiling HuangDepartment of Mathematics and Statistics
    Brock University
    St. Catharines, ON L2S 3A1, Canada
    e-mail
  • Yuanlin LiDepartment of Mathematics and Statistics
    Brock University
    St. Catharines, ON L2S 3A1, Canada
    e-mail

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