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 | References | Similar Articles | Additional Information

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.