On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups

TL;DR

This paper characterizes all minimal product-one sequences of maximal length over Dihedral and Dicyclic groups, and explores their implications for the structure of the monoid of product-one sequences.

Contribution

It provides explicit characterizations of minimal product-one sequences of maximal length over specific non-abelian groups, advancing understanding of their combinatorial properties.

Findings

Explicit characterizations of minimal product-one sequences over Dihedral groups

Explicit characterizations over Dicyclic groups

Analysis of unions of sets of lengths in the monoid of sequences

Abstract

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length $\mathsf D (G)$ over Dihedral and Dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.