On a conjecture of Zhuang and Gao
Volume 171 / 2023
Colloquium Mathematicum 171 (2023), 113-126
MSC: Primary 11B75; Secondary 11P70.
DOI: 10.4064/cm8685-2-2022
Published online: 25 July 2022
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-$1$ subsequence of length $|G|$. In 1961, Erdős, Ginzburg and Ziv proved that $\mathsf E(G)\leq 2|G|-1$ for every finite abelian 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 $G=\langle x, y \mid x^p=y^m=1$, $x^{-1}yx=y^r\rangle $, where $p$ is the smallest prime divisor of $|G|$ and ${\rm gcd}(p(r-1), m)=1$.
