An upper bound for the Davenport constant of finite groups

Let G   be a finite (not necessarily abelian) group and let p=p(G)p=p(G) be the smallest prime number dividing |G||G|. We prove that d(G)⩽|G|p+9p2−10p, where d(G)d(G) denotes the small Davenport constant of G which is defined as the maximal integer ℓ such that there is a sequence over G of length ℓ containing no nonempty one-product subsequence.