On the direct and inverse zero-sum problems over Cn⋊sC2

Abstract

Let Cn be the cyclic group of order n. In this paper, we provide the exact values of some zero-sum constants over G=Cn⋊sC2 where s≢±1(modn), 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 the sequences S1,S2,S3 over G of maximal lengths such that S1 has no short product-one subsequences, S2 has no product-one subsequences of length exp⁡(G), and S3 has no product-one subsequences of length |G|.