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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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When is the commutant of a Bol loop a subloop?
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by Michael K. Kinyon, J. D. Phillips and Petr Vojtěchovský PDF
Trans. Amer. Math. Soc. 360 (2008), 2393-2408 Request permission

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
  • Received by editor(s): January 16, 2006
  • Published electronically: November 27, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 2393-2408
  • MSC (2000): Primary 20N05
  • DOI: https://doi.org/10.1090/S0002-9947-07-04391-7
  • MathSciNet review: 2373318