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arXiv · 1802.02851

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

Jun Seok Oh
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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 $\mathcal B (G)$.

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Jun Seok Oh. 2018-08-29. On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II. https://doi.org/10.1007/s10998-018-00276-9

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