On a class of combinatorial group invariants

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.

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Cite

BibTeX RIS

BibTeX

@misc{fernandes2026class,
  title = {{On a class of combinatorial group invariants}},
  author = {Arthur Fernandes and Claudio Qureshi and Lucas Reis and Sávio Ribas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20516},
  eprint = {2609.20516},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20516},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On a class of combinatorial group invariants
AU  - Arthur Fernandes
AU  - Claudio Qureshi
AU  - Lucas Reis
AU  - Sávio Ribas
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20516
UR  - https://arxiv.org/abs/2609.20516
AB  - 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.
ER  -

Abstract

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