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 constantD(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 nor dihedral 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 is a non-abelian p-group.As a main step in the proof of these bounds, we will also show that D(G)=2q when G is a non-abelian group of order |G|=pq with p and q distinct primes such that p∣q−1.