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 \(\mathsf d (G)\) is the maximal integer \(\ell\) such that there is a sequence over \(G\) of length \(\ell\) which has no nontrivial, product-one subsequence. The large Davenport constant \(\mathsf D (G)\) 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 \(\mathsf d(G)+1\leq \mathsf D(G)\), and if \(G\) is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Now suppose \(G\) has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that \(\mathsf d(G)=\frac12|G|\) if \(G\) is non-cyclic, and \(\mathsf d(G)=|G|-1\) if \(G\) is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that \(\mathsf D(G)=\mathsf d(G)+|G'|\), where \(G'=[G,G]\leq G\) is the commutator subgroup of \(G\).