  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.
