Some zero-sum problems over $\langle x,y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle$

TL;DR

This paper determines exact zero-sum constants for a specific class of groups defined by particular relations, confirming conjectures and solving inverse problems for certain cases.

Contribution

It provides explicit values for zero-sum constants over the group G and verifies conjectures, advancing understanding of zero-sum problems in this group class.

Findings

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

Confirmation of Gao's and Zhuang-Gao's Conjectures for group G.

Solutions to inverse problems when n is divisible by 4.

Abstract

Let $n \ge 8$ be even, and let $G = \langle x, y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle$, where $s^2 \equiv 1 \pmod n$ and $s \not\equiv \pm1 \pmod n$. In this paper, we provide the precise values of some zero-sum constants over $G$, namely the small Davenport constant, $\eta$-constant, Gao constant, and Erd\H os-Ginzburg-Ziv constant. In particular, the Gao's and Zhuang-Gao's Conjectures hold for $G$. We also solve the associated inverse problems when $n \equiv 0 \pmod 4$.