On a conjecture of Zhuang and Gao

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 $\mathsf{E}(G)\le 2|G|-1$ 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 $G=⟨x,y|x^p=y^m=1,x^{-1}yx=y^r⟩$, where $p$ is the smallest prime divisor of $|G|$ and $\mbox{gcd}(p(r-1), m)=1$.

13 pages, main result and some technical lemmas have been used in arXiv:2107.06198