$C_n \rtimes_s C_2$上的极值无积1序列
Extremal product-one free sequences over $C_n \rtimes_s C_2$
摘要：
设$G$是一个有限群，可乘地写成. $G$的小达文波特常数是最大正整数${\sf d}(G)$，使得存在长度为${\sf d}(G)$的序列$S$，其中$S$的每个子序列都是无积1的。令$s^2 \equiv 1 \pmod n$，其中$s \not\equiv \pm1 \pmod n$ .已经证明了${\sf d}(C_n \rtimes_s C_2) = n$（参见[Zhuang，Gao; Europ. J. Combin。26（2005），1053-1059]）。本文确定了$C_n \rtimes_s C_2$上所有长度为$n$的无积1序列.它完成了$C_n \rtimes_s C_2$型群上所有无积1序列的分类，包括拟二面体群和模极大循环群。
英文摘要：
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.
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