The catenary degree of monoids of product-one sequences

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 $S\in \mathcal{B}(G)$ 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.