On the inverse problem of the revised Narkiewicz constant for finite abelian groups
Volume 176 / 2024
Colloquium Mathematicum 176 (2024), 207-217
MSC: Primary 11B75; Secondary 11B30
DOI: 10.4064/cm9413-10-2024
Published online: 27 November 2024
Abstract
For a finite abelian group $G$, let $\eta ^N(G)$ be the smallest integer $t$ such that every sequence $S=g_1\cdot \ldots \cdot g_{t}$ over $G\setminus \{0\}$ of length $t$ has two short zero-sum subsequences $T_1=\prod _{i\in I}g_i$ and $T_2=\prod _{j\in J}g_j$ such that $\prod _{k\in I\cap J}g_k$ is not zero-sum, where $I,J$ are distinct subsets of $[1,t]$. This invariant has close connection with Narkiewicz constant and significant applications in factorization theory. In this paper, we determine the structure of $S$ over $G=C_{n}\oplus C_{nm}$ for $n\in \{2,3\}$, $m\geq 2$ if $|S|=\eta ^N(G)-1$ and $S$ has no such subsequences. Furthermore, we provide the exact value of $\eta ^N(G)$ for these groups.
