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

Abstract

Let $G$ be a finite group and $G^{\prime }$ 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 $|G^{\prime }|\le 2$ if and only if $\mathcal{B}(G)$ is seminormal, and we study sets of lengths in $\mathcal{B}(G)$.