Results in Mathematics

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

On K-Loops Of Finite Order

  • Alexander Kreuzer
  • Heinrich Wefelscheid


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.


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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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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