Extremal product-one free sequences over $C_n{⋊}_s{C}_2$

Abstract

Let $G$ be a finite group multiplicatively written. The small Davenport constant of $G$ is the maximum positive integer $\mathsf{d}(G)$ such that there exists a sequence $S$ of length $\mathsf{d}(G)$ for which every subsequence of $S$ is product-one free. Let $s^2\equiv 1(\mathrm{mod}n)$, where $s\not\equiv \pm 1(\mathrm{mod}n)$. It has been proven that $\mathsf{d}(C_n{⋊}_s{C}_2)=n$ (see Lemma 6 of [Zhuang, Gao; Europ. J. Combin. 26 (2005), 1053-1059]). In this paper, we determine all sequences over $C_n{⋊}_s{C}_2$ of length $n$ which are product-one free. It completes the classification of all product-one free sequences over every group of the form $C_n{⋊}_s{C}_2$, including the quasidihedral groups and the modular maximal-cyclic groups.

9 pages. This is a complete answer to the inverse problem proposed in the old versions of the paper "The $\{1,s\}$-weighted Davenport constant in $C_n^k$". In those old versions, partial answers were given using the bounds of the weighted problem. This complete answer does not use any bound obtained there