The structure of automorphic loops
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- by Michael K. Kinyon, Kenneth Kunen, J. D. Phillips and Petr Vojtěchovský PDF
- Trans. Amer. Math. Soc. 368 (2016), 8901-8927 Request permission
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 $|Q|$, and that $|S|$ divides $|Q|$ 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
- MR Author ID: 267243
- ORCID: 0000-0002-5227-8632
- Email: mkinyon@du.edu
- Kenneth Kunen
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 57306
- MR Author ID: 107920
- Email: kunen@math.wisc.edu
- J. D. Phillips
- Affiliation: Department of Mathematics and Computer Science, Northern Michigan University, Marquette, Michigan 49855
- MR Author ID: 322053
- Email: jophilli@nmu.edu
- Petr Vojtěchovský
- Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
- MR Author ID: 650320
- Email: petr@math.du.edu
- 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.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8901-8927
- MSC (2010): Primary 20N05
- DOI: https://doi.org/10.1090/tran/6622
- MathSciNet review: 3551593