AI 中文总结
本文研究非阿贝尔群的乘积幺序列幺半群的悬链度，刻画了悬链度≤3的有限群，证明某类无限群中特定非阿贝尔群的悬链度为4。
AI 中文摘要
设$G$为乘法表示的有限群。$G$上的序列是取自$G$的有限项集合，允许重复且不计顺序；乘积幺序列是指可对其项排序后，使得它们在$G$中的乘积等于$G$的单位元的序列。所有$G$上乘积幺序列构成的集合$\boldsymbol{\textit{B}}(G)$，以序列的连接为运算，是有限生成的C-幺半群，且是原子的，即每个非单位元都可表示为原子的有限乘积。对$\boldsymbol{\textit{B}}(G)$的研究具有根本重要性，因其组合、代数与算术性质在数学多个分支（最显著的是不变量理论与因子分解理论）中发挥关键作用。尽管在阿贝尔情形下（此时$\boldsymbol{\textit{B}}(G)$是Krull幺半群）$\boldsymbol{\textit{B}}(G)$的算术已被充分理解，但由于非阿贝尔情形存在显著的结构复杂性，相关研究甚少。本文研究非阿贝尔群$G$对应的幺半群$\boldsymbol{\textit{B}}(G)$的算术不变量，重点关注悬链度。幺半群$\boldsymbol{\textit{B}}(G)$的悬链度$\boldsymbol{\textit{c}}(G)$定义为满足以下条件的最小整数$N$：$\boldsymbol{\textit{B}}(G)$中任意元素$S$的两个因子分解，可通过一系列因子分解连接，其中相邻步骤的差异在于替换至多$N$个原子。本文将算术组合学的方法拓展至非阿贝尔情形，明确刻画了所有悬链度不超过3的有限群，并研究了一类无限有限群，其对应的乘积幺序列幺半群是半正规的，且具有良好的算术结构；进一步证明该类中的一个特定非阿贝尔群的悬链度为4。
英文摘要
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.
