关于 Davenport 常数的代数方法

对于有限群 $G,$，$\mathsf{D}(G)$ 定义为最小正整数 $k$，使得对于长度为 $k$ 的每个序列 $S=g_1\cdot g_2\cdot \dotsc\cdot g_k$ over $G$，存在 $1 \le i_1 < i_2 <\cdots< i_m \le k$ 使得 $g_{i_1}g_{i_2}\cdots g_{i_m}=1,$ 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.