On product-one sequences over dihedral groups

TL;DR

This paper investigates the algebraic and combinatorial properties of product-one sequences over dihedral groups, revealing their extremal characteristics and structure within the framework of finitely generated C-monoids.

Contribution

It provides a detailed analysis of the monoid of product-one sequences over dihedral groups using arithmetic combinatorics, highlighting their extremal properties and structural features.

Findings

Characterization of extremal properties of product-one sequences

Structural insights into the monoid of sequences over dihedral groups

Application of arithmetic combinatorics methods

Abstract

Let $G$ be a finite group. A sequence over $G$ means a finite sequence of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over $G$ (with concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This article provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.