On minimal product-one sequences of maximal length over the non-abelian group of order $pq$
Danilo Vilela Avelar, Fabio Enrique Brochero Martínez, Sávio Ribas
TL;DR
The paper addresses the inverse problem for the large Davenport constant over the nonabelian group $\mathcal{G}=C_q\rtimes C_p$ with odd primes and $p\mid q-1$, providing a complete characterization of minimal product-one sequences of maximal length $|S|=2q$ and the explicit atom $S = y^{[q-1]} x y^{[q-1]} x^{p-1} y^{s^{p-1}+1}$. The approach adapts Grynkiewicz’s framework for nonabelian groups, leveraging product-set inequalities and a robust set of lemmas to constrain the structure of atoms and deduce the exact form. As applications, the authors determine the $k$-th elasticity and the unions of sets of lengths for the product-one monoid ${\mathcal B}(\mathcal{G})$, illustrating how nonabelian zero-sum phenomena govern factorization invariants. These results deepen the connection between zero-sum theory in nonabelian groups and factorization theory in Krull-type monoids, providing precise arithmetic information for the group of order $pq$.
Abstract
Let $G$ be a finite group. A sequence over $G$ is a finite multiset of elements of $G$, and it is called product-one if its terms can be ordered so that their product is the identity of $G$. The large Davenport constant $\D(G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence that cannot be partitioned into two nontrivial product-one subsequences. Let $p,q$ be odd prime numbers with $p \mid q-1$ and let $C_q \rtimes C_p$ denote the non-abelian group of order $pq$. It is known that $\D(C_q \rtimes C_p) = 2q$. In this paper, we describe all minimal product-one sequences of length $2q$ over $C_q \rtimes C_p$. As an application, we further investigate the $k$-th elasticity (and, consequently, the union of sets containing $k$) of the monoid of product-one sequences over these groups.