On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II

TL;DR

This paper investigates the algebraic structure of the monoid of product-one sequences over finite groups, characterizing when its class semigroup is Clifford and analyzing sets of lengths within this monoid.

Contribution

It characterizes the class semigroup as Clifford precisely when the commutator subgroup has size at most two, and explores the properties of sets of lengths in the monoid.

Findings

Class semigroup is Clifford iff |G'| ≤ 2.

The monoid is seminormal iff the class semigroup is Clifford.

Analysis of sets of lengths in the monoid.

Abstract

Let $G$ be a finite group and $G'$ its commutator subgroup. 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 monoid $\mathcal B (G)$ of all product-one sequences over $G$ is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if $G$ is abelian (equivalently, $\mathcal B (G)$ is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if $|G'| \le 2$ if and only if $\mathcal B (G)$ is seminormal, and we study sets of lengths in $\mathcal B (G)$.