Extremal product-one free sequences in $C_q{⋊}_s{C}_m$

Abstract

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $d$ such that every sequence of $G$ with $d$ elements has a non-empty subsequence with product $1$. Let $C_n\simeq {\mathbb{Z}}_n$ be the cyclic group of order $n$. Bass (2007) showed that the Davenport constant of the metacyclic group $C_q{⋊}_s{C}_m$, where $q$ is a prime number and ${\text{ord}}_q(s)=m\ge 2$, is $m+q-1$. In this paper, we explicit the form of all sequences $S$ of $C_q{⋊}_s{C}_m$, with $q+m-2$ elements, that are free of product-$1$ subsequences.