Skip to main content
SpringerLink
Log in

Classification of highly symmetrical translation loops of order \(2p, p\) prime

  • Original Paper
  • Published:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Aims and scope Submit manuscript

Abstract

Using tools from algebraic graph theory, we prove that for every odd prime \(p\) there is just one proper loop of order \(2p\) such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order \(4p^2\). Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods.

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

  • Barlotti, A., Strambach, K.: The geometry of binary systems. Adv. Math. 49(1), 1–105 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  • Belousov, V.D.: Foundations of the Theory of Quasigroups and Loops, Moscow; 1967 (in Russian).

  • Biggs, N.L.: Algebraic Graph Theory, Cambridge Tracts in Mathematics, No. 67, Cambridge University Press, Cambridge (1974).

  • Bose, R.C.: Strongly regular graphs, partial geometries, and partially balanced designs. Pac. J. Math. 13, 389–419 (1963)

    Article  MATH  Google Scholar 

  • Bruck, R.H.: Finite nets, I.: Numerical invariants. Can. J. Math. 3, 94–107 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  • Bruck, R.H.: A survey of binary systems. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 20. Reihe: Gruppentheorie, Springer, Berlin, viii \(+\) 185 p (1958).

  • Bruck, R.H.: Finite nets. II. Uniqueness and imbedding. Pac. J. Math. 13, 421–457 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  • Bryant, B.F., Schneider, H.: Principal loop-isotopes of quasigroups. Can. J. Math. 18, 120–125 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  • Chein, O., Pflugfelder, H.O.: The smallest Moufang loop. Archiv der Mathematik 22, 573–576 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  • Dietrich, H., Eick, B.: On the groups of cube-free order. J. Algebra 292(1), 122–137 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Dènes, J., Keedwell, A.D.: Latin Squares and Their Application, 3.1.1, p. 130. Academic press, New York (1974)

  • Drisko, A.A.: Loops with transitive automorphisms. J. Algebra 184, 213–229 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Enomoto, H.: Strongly regular graphs and finite permutation groups of rank 3. J. Math. Kyoto Univ. 11, 381–397 (1971)

    MATH  MathSciNet  Google Scholar 

  • Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (1982)

    Google Scholar 

  • Goodaire, E.G., Robinson, D.A.: A class of loops which are isomorphic to all loop isotopes. Can. J. Math. 34, 662–672 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Heinze, A.: Applications of Schur Rings in Algebraic Combinatorics, Ph.D Thesis, University of Oldenburg (2001)

  • Heinze, A., Klin, M.: Loops, Latin squares and strongly regular graphs: an algorithmic approach via algebraic combinatorics. In: Klin, M., et al. (ed.) Algorithmic Algebraic Combinatorics and Grobner Bases, pp. 3–65. Springer, Berlin (2009)

  • Hestenes, M., Higman, D.G.: Rank 3 subgroups and strongly regular graphs. In: Computers in Number Theory and Algebra (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1970), pp. 141–159. SIAM-AMS Proc., vol. IV, American Mathematical Society, Providence (1971).

  • Jones, G.A., Soomro, K.D.: On a theorem of Wielandt concerning simply primitive groups. Math. Proc. Camb. Philos. Soc. 92, 419–423 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Kiechle, H.: Theory of \(K\)-Loops, Lecture Notes in Mathematics, 1778. Springer, Berlin (2002)

    Google Scholar 

  • Klin, M., Kovacs, M.: Automorphism groups of rational circulant graphs. Electron. J. Combin. 19, P35 (2012)

    MathSciNet  Google Scholar 

  • Klin, M., Meszka, M., Reichard, S., Rosa, A.: The smallest non-rank 3 strongly regular graphs which satisfy the 4-vertex condition. Bayreuth. Math. Schr. No. 74, 145–205 (2005)

    Google Scholar 

  • Kunen, K.: \(G\)-loops and permutation groups. J. Algebra 220(2), 694–708 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Kunen, K.: The structure of conjugacy closed loops. Trans. Am. Math. Soc. 352(6), 2889–2911 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, H.-L.: On groups of order \(p^2q\) and \(p^2q^2\). Tamkang J. Math. 5, 167–190 (1974)

    MATH  MathSciNet  Google Scholar 

  • Marušič, D.: Vertex transitive graphs and digraphs of order \(p^k\), in (Burnaby, B.C., 1982): Cycles in graphs. North-Holland Math. Stud., vol. 115, pp. 115–128. North-Holland, Amsterdam (1985)

  • McKay, B.D.: nauty User’s Guide (Version 1.5) Technical Report TR-CS-90-02. Computer Science Department, Australian National University (1990)

  • McKay, B.D., Meynert, A., Myrvold, W.: Small Latin squares, quasigroups, and loops. J. Combin. Des. 15(2), 98–119 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Moorhouse, G.E.: Bruck nets, codes, and characters of loops. Des. Codes Cryptogr. 1(1), 7–29 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  • Norton, H.W.: The \(7\times 7\) squares. Ann. Eugen. 9, 269–307 (1939)

    Article  MathSciNet  Google Scholar 

  • Pflugfelder, H.O.: Quasigroups and Loops. Introduction. Heidermann, Berlin (1990)

    MATH  Google Scholar 

  • Robinson, D.A.: The Bryant-Schneider group of a loop. Ann. de la Soc. Sci. de Bruxelles 94, 69–81 (1980)

    Google Scholar 

  • Schönhardt, E.: Über Lateinische Quadrate und Unionen. J. Reine Angew. Math. 163, 183–229 (1930)

    MATH  Google Scholar 

  • Schönert, M., et al.: GAP—Groups, Algorithms, and Programming—version 3, release 4, patch level 4. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen (1997)

    Google Scholar 

  • Soicher, L.H.: GRAPE: a system for computing with graphs and groups. In: Finkelstein, L., Kantor, W.M. (eds.) Groups and Computation, vol. 11 DIMACS Series in Discrete Math. and Theor. Comput. Sci., pp. 287–291, Amer. Math. Soc. (1993)

  • Sprague, A.P.: Translation nets. Mitt. Math. Sem. Giessen No.157, 46–68 (1982)

  • Wielandt, H.: Finite Permutation Groups. Academic Press, New York (1964)

    MATH  Google Scholar 

  • Wielandt, H.: Permutation groups through invariant relations and invariant functions. Lecture Notes. The Ohio State University, Columbus (1969)

    Google Scholar 

  • Wilson, E.L.: A class of loops with the isotopy-isomorphy property. Can. J. Math. 18, 589–592 (1966)

    Article  MATH  Google Scholar 

  • Wilson Jr, R.L.: Quasidirect products of quasigroups. Commun. Algebra 3, 835–850 (1975)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to Aiso Heinze, Gareth Jones, Yoav Segev, and Matan Ziv-Av for providing helpful and insightful comments.

Author information

Authors and Affiliations

  1. Ben-Gurion University of the Negev, Beersheba, 84105, Israel

    Mikhail Klin & Nimrod Kriger

  2. Villanova University, Villanova, PA, 19085, USA

    Andrew Woldar

Authors
  1. Mikhail Klin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nimrod Kriger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew Woldar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew Woldar.

Additional information

M. Klin and A. Woldar were supported by a Villanova University Faculty Research Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Klin, M., Kriger, N. & Woldar, A. Classification of highly symmetrical translation loops of order \(2p, p\) prime. Beitr Algebra Geom 55, 253–276 (2014). https://doi.org/10.1007/s13366-012-0132-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-012-0132-4

Keywords

Mathematical Subject Classification

Search

Navigation