Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 5 min ago
PreprintCC BY-NC-SA 4.0arXiv

Quandles from group actions and a Cayley-type embedding theorem

  • Ryoya Kai
Open at sourcePDFDOICite

Abstract

A quandle is an algebraic system that can be regarded as a generalization of the conjugation operation in groups. We study a quandle construction associated with group actions and determine its structural properties, including its inner automorphism group, connected components, and subquandles. As a principal application, we establish a Cayley-type embedding theorem for finite quandles. Applying the construction to the natural action of the symmetric group, we obtain, for each n, a single quandle into which every quandle of cardinality n embeds.

Related

  1. A Lipschitz refinement of the Bebutov-Kakutani dynamical embedding theorem for connected Lie group actions
    Yonatan Gutman, Qiang Huo, Kangbo Ouyang
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  2. Generic Sharp Shift Embedding for Amenable Group Actions with the Uniform Rokhlin Property and Comparison
    Yonatan Gutman, Chunlin Liu
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  3. Lie Quandles, Leibniz Racks and Noether's First Theorem
    Mohamed Elhamdadi, Bryce Virgin
    Symmetry, Integrability and Geometry: Methods and ApplicationsPeer-reviewedCC BY-SA 4.0Full textAlgebra & number theoryCrossref
  4. Distinguishing invariants of Cayley Graphs, Hamming graphs, and wreath product actions
    Amitayu Banerjee
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  5. Composing group automorphisms with colour-preserving automorphisms of Cayley graphs
    Dave Witte Morris
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{kai2026quandles,
  title = {{Quandles from group actions and a Cayley-type embedding theorem}},
  author = {Ryoya Kai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17438},
  eprint = {2609.17438},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17438},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Quandles from group actions and a Cayley-type embedding theorem
AU  - Ryoya Kai
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.17438
UR  - https://arxiv.org/abs/2609.17438
AB  - A quandle is an algebraic system that can be regarded as a generalization of the conjugation operation in groups. We study a quandle construction associated with group actions and determine its structural properties, including its inner automorphism group, connected components, and subquandles. As a principal application, we establish a Cayley-type embedding theorem for finite quandles. Applying the construction to the natural action of the symmetric group, we obtain, for each n, a single quandle into which every quandle of cardinality n embeds.
ER  -