发表机构
Universidade Federal de Minas Gerais; Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação; Universidade Federal de Ouro Preto（米纳斯吉拉斯联邦大学; 圣保罗大学数学与计算科学学院; 欧罗普雷托联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限群组合结构（如幂图）诱导的群不变量及其等价关系划分，发现多数划分重合，并构建通用框架，提出开放问题。
AI 中文摘要
在本文中，我们探讨了由与有限群相关的组合结构（包括幂映射的函数图和已被充分研究的幂图）所产生的群不变量。这些不变量在有限群的同构类集合上诱导出等价关系（从而诱导出划分），我们按从最细到最粗的顺序对这些划分进行分类。令人惊讶的是，这些划分中除三个外全部重合；对于幂零群的子类，除两个外全部重合。此外，我们引入了一大类幂零群，并证明在该类中，这些划分除一个外全部一致。我们的证明运用了组合学与数论中的工具和思想，而仅需群论中的基本概念。特别地，我们构建了一个通用框架，该框架可能在类似于本文所考虑的情境中有所助益。最后，我们提出了从我们的结果中涌现出的一些开放问题。
英文摘要
In this paper, we explore group invariants arising from combinatorial structures associated with finite groups, including the functional graphs of power maps and the well-studied power graphs. These invariants induce equivalence relations (and hence partitions) on the set of isomorphism classes of finite groups, which we classify from the finest to the coarsest. Surprisingly, all but three of these partitions turn out to coincide; for the subclass of nilpotent groups, all but two coincide. Furthermore, we introduce a broad class of nilpotent groups and show that, within this class, all but one of these partitions agree. Our proofs draw on tools and ideas from Combinatorics and Number Theory, while requiring only basic notions from Group Theory. In particular, we construct a general framework that may prove useful in contexts similar to those considered in this paper. Finally, we propose some open questions that emerge from our results.
Comments25 pages; accepted for publication in Algebraic Combinatorics
