On left conjugacy closed loops with a nucleus of index two



A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and middle nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity, but that is associated with the level of commutativity of G (amongst others, the centre of G has to be nontrivial). On the other hand, for each m ≥ 2 one can construct an LCC loop Q of order 2m in such a way that its left nucleus is trivial, and the right nucleus if of order m. If Q is involutorial, then it is a Bol loop.

2000 Mathematics Subject Classification

Primary 20N05 Secondary 08A05 

Key words and phrases

Left conjugacy closed loop nucleus 


  1. [1]
    A. S. Basarab, Klass LK-lup.Matematicheskie issledovanija 120 (1991), 3–7.MathSciNetGoogle Scholar
  2. [2]
    V. D. Belousov,Osnovy teorii kvazigrupp i lup. Nauka, Moskva, 1967.Google Scholar
  3. [3]
    R. H. BrÜck,ASurvey of Binary Systems. Springer-Verlag, 1971.Google Scholar
  4. [4]
    A. Drápal, On multiplication groups of left conjugacy closed loops.Comment. Math. Univ. Carolinae 45 (2004), 223–236.MATHGoogle Scholar
  5. [5]
    E. G. Goodaire andD. A. Robinson, A class of loops which are isomorphic to all loop isotopes.Canad. J. Math. 34 (1982), 662–672.MATHMathSciNetGoogle Scholar
  6. [6]
    —, Semi-direct products and Bol loops.Demonstratio Math. 27 (1994), 573–588.MATHMathSciNetGoogle Scholar
  7. [7]
    H. Kiechle andG. P. Nagy, On the extension of involutorial Bol loops.Abh. Math. Sem. Univ. Hamburg 72 (2002), 235–250.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    M. K. Kinyon, K. Kunen andJ. D. Phillips, Diassociativity in conjugacy closed loops.Communications in Algebra 32 (2004), 767–786.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    G. P. Nagy, Group invariants of certain Burn loop classes.Bull. Belg. Math. Soc. 5 (1998), 403–415.MATHGoogle Scholar
  10. [10]
    P. Nag andK. Strambach, Loops as invariant sections in groups, and their geometry.Canad. J. Math. 46 (1994), 1027–1056.MathSciNetGoogle Scholar

Copyright information

© Mathematische Seminar 2004

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Rep.

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