On Bass' conjecture of the small Davenport constant

TL;DR

We address Bass's conjecture on the small Davenport constant for the metacyclic group $G_{m,n}^*$ under the hypothesis that ${\rm ord}_q s=m$ for every prime divisor $q$ of $n$. We prove $\mathsf d(G_{m,n}^*)=m+n-2$ and give a full inverse classification of extremal product-one free sequences, with explicit forms; the exceptional case $G_{m,n}^*\cong G_{2,3}$ yields two special extremal types. The results extend prior work of Bass (2007), Mart\i\nez and Ribas (2019), and Qu and Li (2022), and the proof combines projection to $G/N\cong C_m$, Kneser-type bounds, and stabilizer arguments. This contributes to non-abelian zero-sum theory and clarifies the structure of product-one sequences in metacyclic groups.

Abstract

Let $G$ be a finite group. The small Davenport constant $\mathsf d(G)$ of $G$ is the maximal integer $\ell$ such that there is a sequence of length $\ell$ over $G$ which has no nonempty product-one subsequence. In 2007, Bass conjectured that $\mathsf d(G_{m,n})=m+n-2$, where $G_{m,n}=\langle x, y| x^m=y^n=1, x^{-1}yx=y^s\rangle$, and $s$ has order $m$ modulo $n$. In this paper, we confirm the conjecture for any group $G_{m,n}$ with additional conditions that $s$ has order $m$ modulo $q$, for every prime divisor $q$ of $n$. Moreover, we solve the associated inverse problem characterizing the structure of any product-one free sequence with extremal length $\mathsf d(G_{m,n})$. Our results generalize some obtained theorems on this problem.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof