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

2018/02/03 by Jun Seok Oh, Oh, Jun Seok
Computer Science · Mathematics · #13A50 #20D60 #20M13 #Advanced Algebra and Logic #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1802.00991

openalex publication_date 2018/02/03 · openalex created_date 2022/10/02 · openalex updated_date 2026/08/01

Abstract

Let G be a finite group. A finite unordered sequence S = g₁ \boldsymbol⋅ … \boldsymbol⋅ g_ℓ 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) ⊂ \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).

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