Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Institution: Universidade Federal de Minas Gerais
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213592265