On the direct and inverse zero-sum problems over non-split metacyclic group

Abstract

Let $G=〈y,x|y^{2n}=1,x^2=y^n,xy{x}^{-1}=y^{\ell }〉$ be the non-split metacyclic group with ${\ell }^2\equiv 1\left(\mathrm{mod}2n\right)$ and $\ell \not\equiv \pm 1,n+1\left(\mathrm{mod}2n\right)$. In this paper, we obtain the exact values of small Davenport constant $d\left(G\right)$, Gao constant $E\left(G\right)$, η-constant $\eta \left(G\right)$ and Erdős-Ginzburg-Ziv constant $s\left(G\right)$. Additionally, we study the associated inverse problems on $d\left(G\right)$, $E\left(G\right)$, $\eta \left(G\right)$ and $s\left(G\right)$. In 2003, Gao conjectured that $s\left(G\right)=\eta \left(G\right)+\text{exp}\left(G\right)-1$ for any finite group G. In 2005, Gao and Zhuang conjectured that $E\left(G\right)=d\left(G\right)+\left|G\right|$ for any finite group G. As a result, we confirm the two conjectures for non-split metacyclic groups.