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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

When is the commutant of a Bol loop a subloop?


Authors: Michael K. Kinyon, J. D. Phillips and Petr Vojtěchovský
Journal: Trans. Amer. Math. Soc. 360 (2008), 2393-2408
MSC (2000): Primary 20N05
DOI: https://doi.org/10.1090/S0002-9947-07-04391-7
Published electronically: November 27, 2007
MathSciNet review: 2373318
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Abstract: A left Bol loop is a loop satisfying $x(y(xz)) = (x(yx))z$. The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order $2k$, $k$ odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to $3$, the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop $K$ such that $K$ is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order $16$ with a non-subloop commutant.


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Additional Information

Michael K. Kinyon
Affiliation: Department of Mathematics, University of Denver, 2360 S Gaylord Street, Denver, Colorado 80208
MR Author ID: 267243
ORCID: 0000-0002-5227-8632
Email: mkinyon@math.du.edu

J. D. Phillips
Affiliation: Department of Mathematics & Computer Science, Wabash College, Crawfordsville, Indiana 47933
MR Author ID: 322053
Email: phillipj@wabash.edu

Petr Vojtěchovský
Affiliation: Department of Mathematics, University of Denver, 2360 S Gaylord Street, Denver, Colorado 80208
MR Author ID: 650320
Email: petr@math.du.edu

Keywords: Bol loop, commutant, extension of loops
Received by editor(s): January 16, 2006
Published electronically: November 27, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.