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Transactions of the American Mathematical Society

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

 
 

 

The structure of commutative automorphic loops


Authors: Přemysl Jedlička, Michael Kinyon and Petr Vojtěchovský
Journal: Trans. Amer. Math. Soc. 363 (2011), 365-384
MSC (2010): Primary 20N05
DOI: https://doi.org/10.1090/S0002-9947-2010-05088-3
Published electronically: August 16, 2010
MathSciNet review: 2719686
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Abstract: An automorphic loop (or A-loop) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and $(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and $p$ a prime. The loop $Q$ has order a power of $p$ if and only if every element of $Q$ has order a power of $p$. The loop $Q$ decomposes as a direct product of a loop of odd order and a loop of order a power of $2$. If $Q$ is of odd order, it is solvable. If $A$ is a subloop of $Q$, then $|A|$ divides $|Q|$. If $p$ divides $|Q|$, then $Q$ contains an element of order $p$. If there is a finite simple nonassociative commutative A-loop, it is of exponent $2$.


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Additional Information

Přemysl Jedlička
Affiliation: Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences, Kamýcká 129, 165 21 Prague 6-Suchdol, Czech Republic
Email: jedlickap@tf.czu.cz

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

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

Received by editor(s): October 6, 2008
Received by editor(s) in revised form: March 31, 2009
Published electronically: August 16, 2010
Additional Notes: The first author was supported by the Grant Agency of the Czech Republic, grant no. 201/07/P015.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.