On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups
Volume 163 / 2021
Colloquium Mathematicum 163 (2021), 317-332
MSC: Primary 11P70; Secondary 11B75.
DOI: 10.4064/cm8033-12-2019
Published online: 31 August 2020
Abstract
Let $G$ be an additive finite abelian group with exponent $\exp (G)$ and $S$ be a sequence of elements of $G$. Let $\Sigma (S) \subset G$ denote the set of group elements which can be expressed as the sum of a nonempty subsequence of $S$. We say $S$ is zero-sum free if $0 \not \in \Sigma (S)$. Suppose $S$ is a zero-sum free sequence of $G$ of length $|S|=\exp (G)+k$, where $k \in \{0,1\}$. It was proved by F. Sun and W. Gao et al. that $|\Sigma (S)|\geq (k+2)\exp (G)-1$. In this paper, we determine the structure of the zero-sum free sequences $S$ such that $|S|=\exp (G)+k$ and $|\Sigma (S)|=(k+2)\exp (G)-1$ for $k\in \{0,1\}$.
