The Large Davenport Constant II: General Upper Bounds

Let $G$ be a finite group written multiplicatively. By a sequence over $G$, we mean a finite sequence of terms from $G$ which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of $G$. The small Davenport constant $\mathsf d (G)$ is the maximal integer $\ell$ such that there is a sequence over $G$ of length $\ell$ which has no nontrivial, product-one subsequence. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence---this is a product-one sequence which cannot be partitioned into two nontrivial, product-one subsequences. The goal of this paper is to present several upper bounds for $\mathsf D(G)$, including the following: $$\mathsf D(G)\leq {ll} \mathsf d(G)+2|G'|-1, & where $G'=[G,G]\leq G$ is the commutator subgroup; \frac34|G|, & if $G$ is neither cyclic nor dihedral of order $2n$ with $n$ odd; \frac{2}{p}|G|, & if $G$ is noncyclic, where $p$ is the smallest prime divisor of $|G|$; \frac{p^2+2p-2}{p^3}|G|, & if $G$ is a non-abelian $p$-group. As a main step in the proof of these bounds, we will also show that $\mathsf D(G)=2q$ when $G$ is a non-abelian group of order $|G|=pq$ with $p$ and $q$ distinct primes such that $p\mid q-1$.