On the direct and inverse zero-sum problems over Cn ⋊s C2.

Abstract

Let Cn be the cyclic group of order n. In this paper, we provide the exact values of some

zero-sum constants over Cn ⋊s C2 where s 6≡ ±1 (mod n), namely η-constant, Gao constant, and Erdős- 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 Cn ⋊s C2 of maximum length.