On a conjecture of Zhuang and Gao

TL;DR

This paper confirms a conjecture relating the minimal sequence length guaranteeing a product-one subsequence in certain finite groups, extending the Erdős-Ginzburg-Ziv theorem to a specific class of non-abelian groups.

Contribution

It proves Zhuang and Gao's conjecture for a class of non-abelian groups defined by specific relations, expanding understanding of zero-sum problems in group theory.

Findings

Confirmed the conjecture for groups generated by two elements with specific relations.

Extended the Erdős-Ginzburg-Ziv theorem to certain non-abelian groups.

Provided new insights into the structure of product-one subsequences in these groups.

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, Erd\H{o}s, Ginzburg and Ziv proved that $\mathsf E(G)\leq 2|G|-1$ for every finite ablian group $G$ and this result is known as the Erd\H{o}s-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 $G=\langle x, y| x^p=y^m=1, x^{-1}yx=y^r\rangle$, where $p$ is the smallest prime divisor of $|G|$ and $\mbox{gcd}(p(r-1), m)=1$.