An algebraic approach towards a conjecture on the Davenport constant

For a finite group $G,$ $\mathsf{D}(G)$ is defined as the least positive integer $k$ such that for every sequence $S=g_1\text{bdot}{g}_2\text{bdot}\dots \text{bdot}{g}_k$ 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 $G\mathrm{.}$ 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 $\mathsf{D}(G)\le \mathsf{L}(G)$ for a finite $p$-group $G$, where $p$ is a prime and $\mathsf{L}(G)$ is the Loewy length of ${\mathbb{F}}_p[G]\mathrm{.}$ 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.