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arXiv · 2501.03338

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

Sávio Ribas
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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 $n \equiv 0 \pmod 4$.

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Sávio Ribas. 2025-01-06. Some zero-sum problems over $\langle x,y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle$. https://arxiv.org/abs/2501.03338

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