On the small Davenport constant for finite groups

Let G be a multiplicatively written finite group. We denote by d(G) the small Davenport constant of G, that is, the maximal integer ℓ such that there is a sequence of length ℓ over G which has no nontrivial product-one subsequence. Recently, we proved a conjecture on the small Davenport constant that d(G) ≤ ∣G∣/p + p − 2 holds for any finite non-cyclic group G, where p is the smallest prime divisor of ∣G∣ ([28]). In this paper, we improve this upper bound by showing that d(G) ≤ ∣G∣/p − 1 for any group G not containing a cyclic subgroup of index p. Consequently, we obtain that if G is a finite non-cyclic group, then the equality d(G) = ∣G∣/p + p − 2 holds if and only if G has a cyclic subgroup of index p.