On a conjecture of Zhuang and Gao

2021/07/14 by Yongke Qu, Qu, Yongke, Yuanlin Li +1 · 2 citations
Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2107.06969

openalex publication_date 2021/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let G be a multiplicatively written finite group. We denote by \mathsf E(G) the smallest integer t such that every sequence of t elements in G contains a product-one subsequence of length |G|. In 1961, Erdős, Ginzburg and Ziv proved that \mathsf E(G)≤ 2|G|-1 for every finite ablian group G and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that \mathsf E(G)=\mathsf d(G)+|G|, where \mathsf d(G) is the small Davenport constant. In this paper, we confirm the conjecture for the case when G=⟨ x, y| x^p=y^m=1, x^-1yx=y^r⟩, where p is the smallest prime divisor of |G| and gcd(p(r-1), m)=1.

Cited by

• On zero-sum problems over metacyclic groups C_n \rtimes_s C₂
• On minimal product-one sequences of maximal length over the non-abelian group of order pq

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