Skip to main content
SpringerLink
Log in

Automorphism groups of quandles and related groups

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper we study various questions concerning automorphisms of quandles. For a conjugation quandle \(Q=\mathrm{Conj}(G)\) of a group G we determine several subgroups of \(\mathrm{Aut}(Q)\) and find necessary and sufficient conditions for these subgroups to coincide with the whole group \(\mathrm{Aut}(Q)\). In particular, we prove that \(\mathrm{Aut}(\mathrm{Conj}(G))=\mathrm{Z}(G)\rtimes \mathrm{Aut}(G)\) if and only if either \(\mathrm{Z}(G)=1\) or G is one of the groups \(\mathbb {Z}_2\), \(\mathbb {Z}_2^2\) or \(\mathbb {Z}_3\). For a big list of Takasaki quandles T(G) of an abelian group G with 2-torsion we prove that the group of inner automorphisms \(\mathrm{Inn}(T(G))\) is a Coxeter group. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles Q with k-transitive action of \(\mathrm{Aut}(Q)\) for \(k\ge 3\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andruskiewitsch, N., Grana, M.: From racks to pointed Hopf algebras. Adv. Math. 178(2), 177–243 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Björner, A., Brenti, F.: Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)

    MATH  Google Scholar 

  3. Bardakov, V., Dey, P., Singh, M.: Automorphism groups of quandles arising from groups. Monatsh. Math. 184(4), 519–530 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burnside, W.: On the outer isomorphisms of a group. Proc. Lond. Math. Soc. S2–11(1), 40–42 (1913)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carter, J.: A Survey of Quandle Ideas, Introductory Lectures on Knot Theory, Series Knots Everything, vol. 46, pp. 22–53. World Science Publisher, Hackensack (2012)

    Book  Google Scholar 

  6. Carter, J., Elhamdadi, M., Nikiforou, M., Saito, M.: Extensions of quandles and cocycle knot invariants. J. Knot Theory Ramif. 12(6), 725–738 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carter, J., Elhamdadi, M., Saito, M.: Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol 2, 95–135 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carter, J., Kamada, S., Saito, M.: Diagrammatic Computations for Quandles and Cocycle Knot Invariants, Diagrammatic Morphisms and Applications (San Francisco, CA, 2000), vol. 318, pp. 51–74. Contemporary Mathematics American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  9. Clark, W., Elhamdadi, M., Saito, M., Yeatman, T.: Quandle colorings of knots and applications. J. Knot Theory Ramif. 23(6), 1450035 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clark, W., Saito, M.: Algebraic properties of quandle extensions and values of cocycle knot invariants. J. Knot Theory Ramif. 25(14), 1650080 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fox, R.: A Quick Trip Through Knot Theory, Topology of 3-Manifolds, pp. 120–167. Prentice-Hall, Upper Saddle River (1962)

    MATH  Google Scholar 

  12. Garcia Iglesias, A., Vendramin, L.: An explicit description of the second cohomology group of a quandle. Math. Z. 286(3–4), 1041–1063 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gorin, E., Lin, V.: Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids. Math. USSR-Sb. 7(4), 569–596 (1969)

    Article  MATH  Google Scholar 

  14. Hulpke, A., Stanovský, D., Vojtěchovský, P.: Connected quandles and transitive groups. J. Pure Appl. Algebra 220(2), 735–758 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23, 37–65 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kamada, S.: Knot Invariants Derived from Quandles and Racks, Invariants of Knots and 3-manifolds (Kyoto, 2001), vol. 4, pp. 103–117. Geometry and Topology Monographs Geometry and Topology Publication, Coventry (2002)

    MATH  Google Scholar 

  17. McCarron, J.: Small homogeneous quandles. In: ISSAC 2012-Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. ACM, New York, pp. 257–264 (2012)

  18. Matveev, S.: Distributive groupoids in knot theory. Mat. Sb. (N.S.) 119–161(1–9), 78–88 (1982). (Russian)

    MathSciNet  Google Scholar 

  19. Nelson, S.: The combinatorial revolution in knot theory. Notices Am. Math. Soc. 58, 1553–1561 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Nosaka, T.: Central extensions of groups and adjoint groups of quandles. ArXiv:Math/1505.03077

  21. Przytycki, J.: 3-coloring and Other Elementary Invariants of Knots, vol. 42, pp. 275–295. Banach Center Publications, Warszawa (1998)

    MATH  Google Scholar 

  22. Stanovský, D.: A guide to self-distributive quasigroups, or latin quandles. Quasigroups Relat. Syst. 23(1), 91–128 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Takasaki, M.: Abstraction of symmetric transformation. Tohoku Math. J. 49, 145–207 (1942)

    MathSciNet  Google Scholar 

  24. Tamaru, H.: Two-point homogeneous quandles with prime cardinality. J. Math. Soc. Jpn 65(4), 1117–1134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Jente Bosmans from Vrije Universiteit Brussel for several remarks and suggestions on Sect. 9. The results given in Sects. 3, 5, 7, 8, 9 are supported by the Russian Science Foundation Project 16-41-02006 (V. Bardakov) and the DST-RSF Project INT/RUS/RSF/P-2 (M. Singh). The results presented in Sects. 4, 6 are supported by the Research Foundation—Flanders (FWO), Grant 12G0317N (T. Nasybullov). We are very grateful to them.

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk, Russia, 630090

    V. Bardakov

  2. Novosibirsk State University, 2 Pirogova Str., Novosibirsk, Russia, 630090

    V. Bardakov

  3. Katholieke Universiteit Leuven KULAK, 53 E. Sabbelaan, 8500, Kortrijk, Belgium

    T. Nasybullov

  4. Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Mohali, Punjab, 140306, India

    M. Singh

Authors
  1. V. Bardakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Nasybullov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Nasybullov.

Additional information

Communicated by A. Constantin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bardakov, V., Nasybullov, T. & Singh, M. Automorphism groups of quandles and related groups. Monatsh Math 189, 1–21 (2019). https://doi.org/10.1007/s00605-018-1202-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-018-1202-y

Keywords

Mathematics Subject Classification

Search

Navigation