Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Article
  • Cited by 20

  • Access

Loops as Invariant Sections in Groups, and their Geometry

  • Péter T. Nagy (a1) and Karl Strambach (a2)

Abstract

We investigate left conjugacy closed loops which can be given by invariant sections in the group generated by their left translations. These loops are generalizations of the conjugacy closed loops introduced in [13] just as Bol loops generalize Moufang loops. The relations of these loops to common classes of loops are studied. For instance on a connected manifold we construct proper topological left conjugacy closed loops satisfying the left Bol condition but show that any differentiable such loop must be a group. We show that the configurational condition in the 3-net corresponding to an isotopy class of left conjugacy closed loops has the same importance in the geometry of 3-nets as the Reidemeister or the Bol condition.

    •

      • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Loops as Invariant Sections in Groups, and their Geometry
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Loops as Invariant Sections in Groups, and their Geometry
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Loops as Invariant Sections in Groups, and their Geometry
      Available formats
      ×

Copyright

COPYRIGHT: © Canadian Mathematical Society 1994

References

Hide All
1. Albert, A. A., Quasigroups II, Trans. Amer. Math. Soc. 55(1944), 401–419.
2. Barlotti, A. and Strambach, K., The geometry of binary systems, Adv. Math. 49(1983), 1–105.
3. Belousov, V. D., Foundations of the Theory of Quasigroups and Loops (Russian), Nauka, Moscow 1976.
4. Brack, R. H., Contributions to the Theory of Loops, Trans. Amer. Math. Soc. 60(1946), 245–354.
5. Brack, R. H., A Survey of Binary Systems, Ergebnisse der Math. 20, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1958.
6. Brack, R. H. and Paige, L. J., Loops whose inner mappings are automorphisms, Ann. in Math. 63(1956), 308–323.
7. Burn, R. P., Finite Bol loops, Math. Proc. Cambridge Philos. Soc. 84(1978), 377–385.
8. Burn, R. P., Finite Bol loops II, Math. Proc. Cambridge Philos. Soc. 89(1981), 445–455.
9. Chein, O., Examples and methods of construction, in Quasigroups and Loops. In: Theory and Applications,(eds. O. Chein, H.O. Pflugfelder and J.D.H. Smith), Sigma Series in Pure Math. 8, Heldermann-Verlag, Berlin 1990, 27–93.
10. Chiboka, V. O., Conjugacy closed loops of Moufang type, Riv. Mat. Pura Appl. 10(1991), 89–93.
11. Fenyves, F., Extra loops I, Publ. Math. Debrecen 15(1968), 235–238.
12. Fenyves, F., Extra loops II. On loops with identities of Bol-Moufang type, Publ. Math. Debrecen 16(1969), 187–192.
13. Goodaire, E. G. and Robinson, D. A., A class of loops which are isomorphic to all loop isotopes, Canad. J. Math. 34(1982), 662–672.
14. Hofmann, K. H. and Strambach, K., Topological and analytical loops, in Quasigroups and Loops. In: Theory and Applications, (eds. O. Chein, H.O. Pflugfelder and J.D.H. Smith), Sigma Series in Pure Math. 8, Heldermann-Verlag, Berlin, 1990, 205–262.
15. Hofmann, K. H. and Strambach, K., Torsion and curvature of smooth loops, Publ. Math. Debrecen 38(1991), 189–214.
16. Holmes, J. P., Dijferentiable power-associative groupoids, Pacific Math. J. 41(1972), 391–394.
17. Huppert, B., Endliche Gruppen I, Grundlehren der Math. Wiss. 134, Springer-Verlag, Berlin, New York, 1967.
18. Kikkawa, M., Geometry of homogenous Lie loops, Hiroshima Math. J. 5 (1975), 141–178.
19. Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry I, Wiley Interscience Publishers, New York, London, 1963.
20. Nagy, P.T., 3-nets with maximal family of two-dimensional subnets, Abh. Math. Sem. Univ. Hamburg 61(1991), 203–211.
21. Pflugfelder, H.O., Quasigroups and Loops: An Introduction, Heldermann-Verlag, Berlin, 1990.
22. Pickert, G., Projektive Ebenen, Springer-Verlag, Berlin, Heidelberg, New York, Second Edition, 1975.
23. Robinson, K., A note on Bol loops of order 2nk, Aequationes Math. 22(1981), 302–306.
24. Shelekhov, A. M., New closure conditions and some problems in loop theory, Aequationes Math. 41(1991), 79–84.
25. Tits, J., Liesche Gruppen und Algebren, Hochschultext, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

  • Cited by 20

  • Access

Loops as Invariant Sections in Groups, and their Geometry

  • Péter T. Nagy (a1) and Karl Strambach (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed