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Loops as Invariant Sections in Groups, and their Geometry

Published online by Cambridge University Press:  20 November 2018

Péter T. Nagy  and
Karl Strambach
Péter T. Nagy
Affiliation:
Bolyai Institute University of Szeged, Aradi vértanulc tere I H-6720 Szeged, Hungary
Karl Strambach
Affiliation:
Mathematisches Institut der Universität Erlangen-Nürnberg, Bismarckstr. 1½ D-91054 Erlangen, Germany
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Abstract

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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.

Keywords

20N0520B9922E9905B3051A9953C30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 1027 - 1056
Copyright
Copyright © Canadian Mathematical Society 1994

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