Extremal product-one free sequences and $|G|$-product-one free sequences of a metacyclic group

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 abelian group $G$ and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that $\mathsf{E}(G)=\mathsf{d}(G)+|G|$ for every finite group, where $\mathsf{d}(G)$ is the small Davenport constant. Very recently, we confirmed this 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 $\mbox{gcd}(p(r-1), m)=1$. In this paper, we study the associated inverse problems on $\mathsf{d}(G)$ and $\mathsf{E}(G)$. Our main results characterize the structure of any product-one free sequence with extremal length $\mathsf{d}(G)$, and that of any $|G|$-product-one free sequence with extremal length $\mathsf{E}(G)-1$.

25 pages. This paper deals with the inverse problem related to arXiv:2107.06969