The catenary degree of monoids of product-one sequences

2026/08/02 by Jun Seok Oh
Mathematics · #math.GR #math.AC #math.CO #math.NT #msc:11B30 #msc:11P70 #msc:11R27 #msc:13F45 #msc:20D15 #msc:20D60 #msc:20M13

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arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

Let G be a (multiplicatively written) finite group. A sequence over G is a finite collection of terms from G, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose terms can be ordered such that their product in G equals the identity element of G. The set \mathcal B (G) of all product-one sequences over G, endowed with the concatenation of sequences as the operation, is a finitely generated C-monoid; in particular, it is atomic, i.e., every non-unit element can be written as a finite product of atoms. The study of \mathcal B (G) is of fundamental importance, as its combinatorial, algebraic, and arithmetic properties play a crucial role across various branches of mathematics, most notably in invariant theory and factorization theory. While the arithmetic of the monoid \mathcal B (G) is well understood in the abelian setting (in which case \mathcal B (G) is a Krull monoid), little is known in the non-abelian setting because of the substantial structural complexity involved. In this paper, we study the arithmetic invariants of the monoid \mathcal B (G) for non-abelian groups, focusing in particular on the catenary degree. The catenary degree \mathsf c (G) of the monoid \mathcal B (G) is defined as the smallest integer N such that any two factorizations of an element S ∈ \mathcal B (G) can be concatenated by a chain of factorizations in which adjacent steps differ by replacing at most N atoms. Extending the methods from arithmetic combinatorics to the non-abelian setting, we explicitly characterize all finite groups with catenary degree at most 3, and we investigate an infinite class of finite groups whose monoids of product-one sequences are seminormal and possess well-behaved arithmetic structures. Furthermore, we show that a specific non-abelian group in this class has catenary degree 4.

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