The main zero-sum constants over $D_{2n} \times C_2$

TL;DR

This paper determines exact zero-sum constants for the group $D_{2n} imes C_2$, proving conjectures and providing new insights into non-abelian zero-sum problems of higher rank.

Contribution

It computes exact zero-sum constants for $D_{2n} imes C_2$ and confirms conjectures, advancing understanding of non-abelian groups of rank greater than two.

Findings

Exact values of small Davenport, Gao, η, and Erdős-Ginzburg-Ziv constants.

Proof of Gao's and Zhuang-Gao's Conjectures for the group.

First concrete results on zero-sum problems for certain non-abelian groups.

Abstract

Let $C_2$ be the cyclic group of order $2$ and $D_{2n}$ be the dihedral group of order $2n$, where $n$ is even. In this paper, we provide the exact values of some zero-sum constants over $D_{2n} \times C_2$, namely small Davenport constant, Gao constant, $\eta$-constant and Erd\H os-Ginzburg-Ziv constant. As a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for this group. These are the first concrete results on zero-sum problems for a family of non-abelian groups of rank greater than $2$.