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

TL;DR

This work determines exact zero-sum invariants for a specific non-abelian group $G$ given by ${G = <x,y | x^2 = y^{n/2}, y^n = 1, yx = xy^s>}$ with even ${n\ge 8}$ and ${s^2 \equiv 1 \pmod n}$, ${s \not\equiv \pm1}$. It proves ${d(G)=n}$, ${\eta(G)=n+1}$, and, depending on ${n mod 4}$, ${s(G)=2n}$ or ${3n}$ with ${E(G)=3n}$, also establishing Gao's and Zhuang–Gao's conjectures for this family. The paper develops the direct (existence) problems via an inductive approach using normal subgroups and quotient groups, leveraging known results for cyclic, dicyclic, and elementary abelian-like groups to obtain tight bounds. It then fully resolves the inverse problems when ${n \equiv 0 \pmod 4}$, giving explicit characterizations of extremal sequences of lengths ${2n-1}$, ${3n-1}$, and ${n}$ in terms of explicit generators and gcd conditions. The results solidify a non-abelian zero-sum theory for this class and provide a template for handling similar inductive setups in non-abelian groups.

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, $η$-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 $n \equiv 0 \pmod 4$.

Theorems & Definitions (8)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof