On product-one sequences over dihedral groups

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.