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Results in Mathematics

, Volume 25, Issue 1–2, pp 79–102 | Cite as

On K-Loops Of Finite Order

  • Alexander Kreuzer
  • Heinrich Wefelscheid
Article

Abstract

In this note we undertake an axiomatic investigation of K-loops (or gyrogroups, as A.A.Ungar used to name them) and provide new construction methods for finite K-loops. It is shown how, more or less, the axioms are independent from each other. Especially (K6) is independent as A.A. Ungar already had conjectured. We begin with right loops (L,⊕) and add step by step further properties. So the connection between K-loops, Bol-loops, Bruck-loops and the homogeneous loops of Kik-kawa became clear. The smallest examples of proper K-loops possess 8 elements; there are exactly 3 non-isomorphic of these.

At last it is shown that one gets quite naturally a Frobenius-group as a quasidirect product of a K-loop (L,⊕) and a group D of automorphisms of (L,⊕) if D is fixed point free except from 0.

Keywords

Neutral Element Transitive Group Inverse Property Loop Operation Axiomatic Investigation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Bol, G. Gewebe und Gruppen. Math Ann. 114 (1937), 414–431MathSciNetCrossRefGoogle Scholar
  2. [2]
    Bruck, R. H.: A survey of binary systems. Springer — Verlag, Berlin 1958zbMATHGoogle Scholar
  3. [3]
    Bruck, R. H. and Paige, L. J.: Loops whose inner mappings are automorphisms. Annals Math. 63 (1956), 308–323MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Burn, R. P. Finite Bol loops. Math. Proc. Cambridge Philos. Soc. 84 (1978), 377–385MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Chein, O., Pflugfelder, H.O., Smith, J. D. H.: Quasigroups and Loops, Theory and Applications. Heldermann Verlag, Berlin 1990Google Scholar
  6. [6]
    Glauberman, G.: On Loops of Odd Order. J. Algebra 1 (1966), 374–396MathSciNetCrossRefGoogle Scholar
  7. [7]
    Gräter, J.: Letter to the authors, 6 April 1993Google Scholar
  8. [8]
    Im, B.: K-loops and their generalisations. Beiträge zur Geometrie und Algebra 23(1993), TUM-Bericht M 9312, 9–17Google Scholar
  9. [9]
    Karzel, H.: Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom. Abh. Math. Sem. Univ. Hamburg 32 (1968), 191–206MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Karzel, H.: The Lorentz group and the hyperbolic Geometry. Beiträge zur Geometrie und Algebra 24 (1993), TUM-Bericht M 9315, 10–22Google Scholar
  11. [11]
    Karzel, H and Wefelscheid, H.: Groups with ah involutory antiautomorphism and K-loops; Application to Space — Time — World and hyperbolic geometry. Res. Math. 23 (1993), 338–354MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Kepka, T.: A construction of Bruuck loops. Commentationes Math. Univ. Carolinae 25,4 (1984), 591–595.MathSciNetGoogle Scholar
  13. [13]
    Kerby, W.: Infinite sharply multiple transitive groups. Hamburger Mathematische Einzelschriften, Neue Folge, Heft 6. Vandenhoek und Ruprecht, Göttingen 1974Google Scholar
  14. [14]
    Kerby, W. und Wefelscheid, H.: Bemerkungen über Fastbereiche und scharf 2-fach transitive Gruppen. Abh. Math. Sem. Uni. Hamburg 37 (1971), 20–29MathSciNetCrossRefGoogle Scholar
  15. [15]
    Kerby, W. und Wefelscheid, H.: Über eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur. Abh. Math. Sem. Univ. Hamburg 37 (1972), 225–235MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Kerby, W. and Wefelscheid, H.: Conditions of finiteness in sharply 2-transitive groups. Aequat. Math. 8 (1974), 169–172MathSciNetGoogle Scholar
  17. [17]
    Kerby, W. and Wefelscheid, H.: The maximal subnearfield of a neardomain. J. Algebra 28 (1974), 319–325MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Kikkawa, M.: Geometry of homogeneous Lie loops. Hiroshima Math J. 5 (1975), 141–179MathSciNetzbMATHGoogle Scholar
  19. [19]
    Kist, G.: Theorie der verallgemeinerten kinematischen Räume. Beiträge zur Geometrie und Algebra 14, TUM-Bericht M 8611, München 1986Google Scholar
  20. [20]
    Kolb, E. and Kreuzer, A.: Geometry of kinematic K-loops. Preprint.Google Scholar
  21. [21]
    Kreuzer, A.: Beispiele endlicher und unendlicher K-Loops. Res. Math. 23 (1993), 355–362MathSciNetzbMATHGoogle Scholar
  22. [22]
    Kreuzer, A.: K-loops and Brück loops on ℝ × ℝ. J. of Geometry 47 (1993)Google Scholar
  23. [23]
    Kreuzer, A.: Algebraische Struktur der relativistischen Geschwindigkeitsaddition. Beiträge zur Geometrie und Algebra 23 (1993), TUM-Bericht M9312, 31–44Google Scholar
  24. [24]
    Kreuzer, A.: Construction of loops of even order. Beiträge zur Geometrie und Algebra 24 (1993), TUM-Bericht M9315, 10–22Google Scholar
  25. [25]
    Niederreiter, H. and Robinson, K. H.: Bol loops of order pq. Math. Proc. Cambridge Philos. Soc. 89 (1981), 241–256MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Robinson, D. A.: Bol-loops. Trans Amer. Math. Soc. 123 (1966), 341–354MathSciNetCrossRefGoogle Scholar
  27. [27]
    Robinson, K., H.: A note on Bol loops of order 2nk. Aequationes Math. 22 (1981) 302–306MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Sherma, B. L. and Solarin, A. R. T.: On the Classification of Bol loops of order 3p (p>3). Communicationes in Algebra 16(1), (1988), 37–55CrossRefGoogle Scholar
  29. [29]
    Ungar, A., A.: Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett. 1 (1988), 57–89MathSciNetCrossRefGoogle Scholar
  30. [30]
    Ungar, A., A.: Weakly associative groups. Res. Math. 17(1990), 149–168MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Ungar, A., A.: Group-like structure underlying the unit ball in real inner product spaces. Res. Math 18 (1990), 355–364MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Ungar, A.A.: Several letters to the autors (1990–1993)Google Scholar
  33. [33]
    Wähling, H.: Theorie der Fastkörper. Thales Verlag, Essen 1987zbMATHGoogle Scholar
  34. [34]
    Wefelscheid, H.: ZT-subgroups of sharply 3-transitive groups. Proc. Edinburgh Math. Soc. 23 (1980),9–14MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1994

Authors and Affiliations

  • Alexander Kreuzer
    • 1
  • Heinrich Wefelscheid
    • 2
  1. 1.Mathematisches InstitutTechnische Universität MünchenMünchen
  2. 2.Fachbereich 11/ MathematikUniversität DuisburgDuisburg

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