Classification of finite subgroups of generalized cactus groups

Abstract

We classify the finite groups that embed into generalized cactus groups associated with arbitrary Coxeter systems. Every finite subgroup is conjugate to a subgroup of the generalized cactus group associated with a finite standard parabolic subgroup of the underlying Coxeter group. This reduces the classification to explicit lists of finite groups for the finite irreducible Coxeter systems. For types B, D, E, F, and H, we analyze cube stabilizers to obtain these lists. For type I₂(p), we determine the associated generalized cactus group directly from its defining presentation. We explicitly realize every listed group as a subgroup of the corresponding generalized cactus group. When the Coxeter diagram has finitely many vertices, we determine the largest order and all possible orders of finite subgroups. In type A, we prove directly that a finite group embeds into the ordinary cactus group on n strands if and only if it embeds into a Sylow 2-subgroup of the symmetric group on n letters. We also give weighted embedding criteria in types A and B and counterexamples to Genevois's conjecture on finite subgroups of cactus groups.

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BibTeX RIS

BibTeX

@misc{kim2026classification,
  title = {{Classification of finite subgroups of generalized cactus groups}},
  author = {Junseok Kim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27721},
  eprint = {2610.27721},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27721},
}

RIS

TY  - UNPB
TI  - Classification of finite subgroups of generalized cactus groups
AU  - Junseok Kim
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.27721
UR  - https://arxiv.org/abs/2610.27721
AB  - We classify the finite groups that embed into generalized cactus groups associated with arbitrary Coxeter systems. Every finite subgroup is conjugate to a subgroup of the generalized cactus group associated with a finite standard parabolic subgroup of the underlying Coxeter group. This reduces the classification to explicit lists of finite groups for the finite irreducible Coxeter systems. For types B, D, E, F, and H, we analyze cube stabilizers to obtain these lists. For type I_2(p), we determine the associated generalized cactus group directly from its defining presentation. We explicitly realize every listed group as a subgroup of the corresponding generalized cactus group. When the Coxeter diagram has finitely many vertices, we determine the largest order and all possible orders of finite subgroups. In type A, we prove directly that a finite group embeds into the ordinary cactus group on n strands if and only if it embeds into a Sylow 2-subgroup of the symmetric group on n letters. We also give weighted embedding criteria in types A and B and counterexamples to Genevois's conjecture on finite subgroups of cactus groups.
ER  -

Abstract

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