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

Abstract

Let G be a finite group. A finite unordered sequence S=g1 ∙⋯ ∙gℓ of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals 1G, the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid ℱ(G) with basis G, and we study the submonoid ℬ(G)⊂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, ℬ(G) is a well-studied object. In the present paper we focus on nonabelian groups, and we study the class semigroup and the arithmetic of ℬ(G).