The structure of sequences with zero-sum subsequences of the same length on finite abelian groups of rank two
Wanzhen Hui, Xue Li
TL;DR
This work advances the inverse zero-sum problem for rank-two finite abelian groups by giving a complete structural classification of extremal sequences for $\mathrm{disc}(G)-1$ when all nonempty zero-sum subsequences have the same length, for $G= C_n \oplus C_{nm}$ with $n,m \ge 2$. Up to generating-set equivalence with $\operatorname{ord}(g_2)=nm$, such a sequence $S$ must be one of five explicit forms (1)–(5). The proof combines the exact values $\mathsf D(G)=n+nm-1$ and $\mathrm{disc}(G)=\mathsf D(G)+\exp(G)$ with a key lemma on the supports of zero-sum subsequences and a detailed case analysis of the canonical form $ST^{-1}(-\sigma(ST^{-1}))$, yielding the full classification. These results enhance understanding of extremal zero-sum structures and provide concrete templates for further study of inverse disc problems in low-rank abelian groups.
Abstract
Let $G$ be an additive finite abelian group, and let $\mathrm{disc}(G)$ denote the smallest positive integer $t$ with the property that every sequence $S$ over $G$ with length $|S|\geq t $ contains two nonempty zero-sum subsequences of distinct lengths. In recent years, Gao et al. established the exact value of $\mathrm{disc}(G)$ for all finite abelian groups of rank $2$ and resolved the corresponding inverse problem for the group $C_n \oplus C_n$. In this paper, we characterize the structure of sequences $S$ over $G = C_n \oplus C_{nm}$ (where $m\geq 2$) when $|S| = \mathrm{disc}(G)- 1$ and all nonempty zero-sum subsequences of $S$ have the same length.