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The structure of automorphic loops


Authors: Michael K. Kinyon, Kenneth Kunen, J. D. Phillips and Petr Vojtěchovský
Journal: Trans. Amer. Math. Soc. 368 (2016), 8901-8927
MSC (2010): Primary 20N05
DOI: https://doi.org/10.1090/tran/6622
Published electronically: March 21, 2016
MathSciNet review: 3551593
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Abstract: Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops.

We study uniquely $ 2$-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely $ 2$-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop $ Q$ of odd order is solvable and contains an element of order $ p$ for every prime $ p$ dividing $ \vert Q\vert$, and that $ \vert S\vert$ divides $ \vert Q\vert$ for every subloop $ S$ of $ Q$.

There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than $ 2500$. We show that if $ Q$ is a finite simple nonassociative automorphic loop, then the socle of the multiplication group of $ Q$ is not regular. The existence of a finite simple nonassociative automorphic loop remains open.

Let $ p$ be an odd prime. Automorphic loops of order $ p$ or $ p^2$ are groups, but there exist nonassociative automorphic loops of order $ p^3$, some with trivial nucleus (center) and of exponent $ p$. We construct nonassociative ``dihedral'' automorphic loops of order $ 2n$ for every $ n>2$, and show that there are precisely $ p-2$ nonassociative automorphic loops of order $ 2p$, all of them dihedral.


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Additional Information

Michael K. Kinyon
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: mkinyon@du.edu

Kenneth Kunen
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 57306
Email: kunen@math.wisc.edu

J. D. Phillips
Affiliation: Department of Mathematics and Computer Science, Northern Michigan University, Marquette, Michigan 49855
Email: jophilli@nmu.edu

Petr Vojtěchovský
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: petr@math.du.edu

DOI: https://doi.org/10.1090/tran/6622
Keywords: Automorphic loop, inner mapping, Odd Order Theorem, Cauchy Theorem, Lagrange Theorem, solvable loop, Bruck loop, Lie ring, middle nuclear extension, dihedral automorphic loop, simple automorphic loop, primitive group
Received by editor(s): October 4, 2012
Received by editor(s) in revised form: November 26, 2014
Published electronically: March 21, 2016
Additional Notes: The fourth author was partially supported by Simons Foundation Collaboration Grant 210176.
Article copyright: © Copyright 2016 American Mathematical Society