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Construction of Finite Loops of Even Order

  • Alexander Kreuzer
Part of the Mathematics and Its Applications book series (MAIA, volume 336)

Abstract

Using a construction method of [9], we give examples of non-associative loops with additional properties. These are power associative, left alternative loops which satisfy the automorphic inverse property and the left inverse property but not the Bol identity. It will be shown that, for n, k ∈ ℕ, non-isomorphic K-loops (L, ⊕) of order 8kn exist which are also Bruck loops, having commutative subgroups (G, ⊕) and (H, ⊕) of order An and 2k, respectively with L = G ⊕ H.

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References

  1. [1]
    Bol, G., Gewebe und Gruppen. Math, Ann. 114 (1937), 414–431MathSciNetCrossRefGoogle Scholar
  2. [2]
    BRÜck, R. H., A survey of binary systems. Springer-Verlag, Berlin 1958.Google Scholar
  3. [3]
    Chein, O., Pflugfelder, H. O., Smith, J. D. H., Quasigroups and Loops, Theory and Applications. Heldermann Verlag, Berlin 1990.Google Scholar
  4. [4]
    Glauberman, G., On loops of odd order. J. Algebra 1 (1966), 374–396MathSciNetCrossRefGoogle Scholar
  5. [5]
    Karzel, H., Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom. Abh. Math. Sem. Univ. Hamburg 32 (1968), 191–206MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Kepka, T., A construction of Brück loops. Commentationes Math. Univ. Carolinae 25 (1984), 591–595.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Kist, G., Theorie der verallgemeinerten kinematischen Räume. Beiträge zur Geometrie und Algebra 14, TUM-Bericht M 8611, München 1986.Google Scholar
  8. [8]
    Kreuzer, A., Beispiele endlicher und unendlicher K–Loops. Res. Math. 23 (1993), 355–362.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Kreuzer, A. and Wefelscheid, H., On K-loops of fínite order. Res. Math. 25 (1994).Google Scholar
  10. [10]
    Niederreiter, H. and Robinson, K. H., Bol loops of order pq. Math. Proc. Cambridge Philos. Soc. 89 (1981), 241–256.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Robinson, D. A., Bol-loops. Trans. Amer. Math. Soc. 123 (1966), 341–354.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Robinson, K.H., A note on Bol loops of order 2 nk. Aequationes Math. 22(1981)302–306.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Ungar, A. A., Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett. 1 (1988), 57–89.MathSciNetCrossRefGoogle Scholar
  14. [14]
    UNGAR, A. A., Weakly associative groups. Res. Math. 17 (1990), 149–168.MathSciNetzbMATHGoogle Scholar
  15. [15]
    WÄhling, H., Theorie der Fastkörper. Thaies Verlag, Essen 1987.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Alexander Kreuzer
    • 1
  1. 1.Mathematisches InstitutTechnische Universität MünchenMünchenGermany

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