Fabio Enrique Brochero Martínez, Sávio Ribas
Published 2021-07-16Version 1
Let $G$ be a finite group multiplicatively written. The small Davenport constant of $G$ is the maximum positive integer ${\sf d}(G)$ such that there exists a sequence $S$ of length ${\sf d}(G)$ for which every subsequence of $S$ is product-one free. Let $s^2 \equiv 1 \pmod n$, where $s \not\equiv \pm1 \pmod n$. It has been proven that ${\sf d}(C_n \rtimes_s C_2) = n$ (see Lemma 6 of [Zhuang, Gao; Europ. J. Combin. 26 (2005), 1053-1059]). In this paper, we determine all sequences over $C_n \rtimes_s C_2$ of length $n$ which are product-one free. It completes the classification of all product-one free sequences over every group of the form $C_n \rtimes_s C_2$, including the quasidihedral groups and the modular maximal-cyclic groups.
Comments: 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
Extremal product-one free sequences in $C_q \rtimes_s C_m$
Extremal product-one free sequences in Dihedral and Dicyclic groups
The main zero-sum constants over $D_{2n} \times C_2$
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