The large Davenport constant I: Groups with a cyclic, index 2 subgroup

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 constant d(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 factored into two nontrivial, product-one subsequences. It is easily observed that d(G)+1≤D(G), and if G is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Suppose G has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that d(G)=12|G| if G is non-cyclic, and d(G)=|G|−1 if G is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that D(G)=d(G)+|G′|, where G′=[G,G]≤G is the commutator subgroup of G.