An algebraic approach towards a conjecture on the Davenport constant

TL;DR

The paper investigates the ordered Davenport constant $\mathsf{D}(G)$ and its relation to the Loewy length $\mathsf{L}(G)$ for finite $p$-groups, testing Dimitrov's conjecture $\mathsf{D}(G)=\mathsf{L}(G)$ in new non-abelian classes. It computes exact values of $\mathsf{D}(G)$ and $\mathsf{L}(G)$ for metacyclic-type groups $G_1(\alpha,\beta,\gamma)$, $G_2(\alpha,\beta)$, and $G_3(\alpha,\beta,\sigma)$, establishing equality $\mathsf{D}(G)=\mathsf{L}(G)$ in these cases and deriving explicit formulas such as $\mathsf{L}(G_1(\alpha,\beta,\gamma))=p^{\alpha}+p^{\beta}+2p^{\gamma}-3$ and $\mathsf{L}(G_2(\alpha,\beta))=\mathsf{D}(G_2(\alpha,\beta))=p^{\alpha}+p^{\beta}-1$. The authors also determine $\mathsf{D}(G)$ for dicyclic, semi-dihedral, and related groups, refining known upper bounds for the small Davenport constant and showing that $\mathsf{D}(G)=\mathsf{d}(G)+1$ in several instances. The methods combine power-subgroup structure, the $M$-series (Brauer–Jennings–Zassenhaus) and quadratic non-residue arguments to obtain tight bounds and exact values, advancing the non-abelian zero-sum theory and supporting Dimitrov's conjecture in broader families.

Abstract

For a finite group $G,$ $\mathsf{D}(G)$ is defined as the least positive integer $k$ such that for every sequence $S=g_1\bdot g_2\bdot \dotsc \bdot g_k$ of length $k$ over $G$, there exist $1 \le i_1 < i_2 <\cdots < i_m \le k $ such that $g_{i_1}g_{i_2}\cdots g_{i_m}=1,$ where $1$ is the identity element of $G.$ The small Davenport constant $\mathsf{d}(G)$ is the maximal positive integer $k$ such that there is a sequence of length $k$ over $G$ which has no non-trivial product-one subsequence. In 2004, Dimitrov proved that $\mathsf{D}(G)\leq \mathsf{L}(G)$ for a finite $p$-group $G$, where $p$ is a prime and $\mathsf{L}(G)$ is the Loewy length of $\mathbb{F}_p[G].$ He conjectured that the equality holds for all finite $p$-groups. In this article, we compute $\mathsf{D}(G)$ for certain classes of finite non-abelian $p$-groups, including metacyclic groups, and show that the conjecture is true by determining the precise value of $\mathsf{L}(G)$. As a consequence, we refine an upper bound on $\mathsf{d}(G)$ recently given by Qu, Li and Teeuwsen, and prove that for specific classes of groups $\mathsf{D}(G)=\mathsf{d}(G)+1$. We also evaluate $\mathsf{D}(G)$ for finite dicyclic, semi-dihedral and other groups.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Remark 1.1
• Corollary 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 2.1