On the algebraic and arithmetic structure of the monoid of product-one sequences

TL;DR

This paper investigates the algebraic and arithmetic properties of the monoid of product-one sequences over finite groups, especially focusing on non-abelian groups, extending known results from abelian cases.

Contribution

It analyzes the class semigroup and arithmetic structure of the monoid of product-one sequences for non-abelian finite groups, a less-explored area.

Findings

Characterization of the class semigroup for non-abelian groups

Analysis of the arithmetic properties of the monoid

Extension of known results from abelian to non-abelian groups

Abstract

Let $G$ be a finite group. A finite unordered sequence $S = g_1 \boldsymbol{\cdot} \ldots \boldsymbol{\cdot} g_{\ell}$ of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals $1_G$, the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid $\mathcal F (G)$ with basis $G$, and we study the submonoid $\mathcal B (G) \subset \mathcal F (G)$ of all product-one sequences. This is a finitely generated C-monoid, which is a Krull monoid if and only if $G$ is abelian. In case of abelian groups, $\mathcal B (G)$ is a well-studied object. In the present paper we focus on non-abelian groups, and we study the class semigroup and the arithmetic of $\mathcal B (G)$.