On product-one sequences over subsets of groups

TL;DR

This paper investigates the algebraic and arithmetic properties of monoids formed by product-one sequences over finite and infinite subsets of groups, with a focus on infinite dihedral groups, advancing understanding of their structure.

Contribution

It introduces new analyses of monoids of product-one sequences over various group subsets, especially emphasizing infinite dihedral groups, and explores their algebraic and arithmetic properties.

Findings

Characterization of monoids of product-one sequences over finite subsets

Analysis of algebraic properties of these monoids

Insights into the structure of infinite dihedral groups

Abstract

Let $G$ be a group and $G_0 \subseteq G$ be a subset. A sequence over $G_0$ means a finite sequence of terms from $G_0$, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. We study algebraic and arithmetic properties of monoids of product-one sequences over finite subsets of $G$ and over the whole group $G$, with a special emphasis on infinite dihedral groups.