# The large davenport constant II: General upper bounds

## Abstract

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 constantd(G) is the maximal integer ℓ such that there is a sequence over G of length ℓ which has no nontrivial, product-one subsequence. The large Davenport constant 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 D(G), including the following: D(G) ≤ d(G)+2|G'|-1, where G'=[G,G] ≤ G is the commutator subgroup; 34|G|, if G is neither cyclic nordihedral of order 2n with n odd; 2p|G|, if G is noncyclic, where p is the smallest prime divisor of |G|; p2+2p-2p3|G|, if G isanon-abelian p-group. As a main step in the proof of these bounds, we will also show that D(G) = 2 q when G is a non-abelian group of order |G| = pq with p and q distinct primes such that pdivides q-1. © 2013 Elsevier B.V.

## Publication Title

Journal of Pure and Applied Algebra

## Recommended Citation

Grynkiewicz, D.
(2013). The large davenport constant II: General upper bounds.* Journal of Pure and Applied Algebra**, 217* (12), 2221-2246.
https://doi.org/10.1016/j.jpaa.2013.03.002