On the direct and inverse zero-sum problems over $C_n \rtimes_s C_2$

Abstract: Let $C_n$ be the cyclic group of order $n$. In this paper, we provide the exact values of some zero-sum constants over $C_n \rtimes_s C_2$ where $s \not\equiv \pm1 \pmod n$, namely $\eta$-constant, Gao constant, and Erd\H{o}s-Ginzburg-Ziv constant (the latter for all but a "small" family of cases). As a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for groups of this form. We also solve the associated inverse problems by characterizing the structure of product-one free sequences over $C_n \rtimes_s C_2$ of maximum length.