An upper bound for Davenport constant of finite groups

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)leq frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as the maximal integer $ell$ such that there is a sequence over $G$ of length $ell$ contains no nonempty one-product subsequence.

On the Index of Sequences over Cyclic Groups

Let $G$ be a finite cyclic group of order $n ge 2$. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdot ... cdot (n_lg)$ where $gin G$ and $n_1,..., n_l in [1,ord(g)]$, and the index $ind (S)$ of $S$ is defined as the minimum of $(n_1+ ... + n_l)/ord (g)$ over all $g in G$ with $ord (g) = n$. In this paper we prove that a sequence $S$ over $G$ of length $|S| = n$ having an element with multiplicity at least $frac{n}{2}$ has a subsequence $T$ with $ind (T) = 1$, and if the group order $n$ is a prime, then the assumption on the multiplicity can be relaxed to $frac{n-2}{10}$. On the other hand, if $n=4k+2$ with $k ge 5$, we provide an example of a sequence $S$ having length $|S| > n$ and an element with multiplicity $frac{n}{2}-1$ which has no subsequence $T$ with $ind (T) = 1$. This disproves a conjecture given twenty years ago by Lemke and Kleitman.