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Solvability of commutative automorphic loops


Authors: Alexander Grishkov, Michael Kinyon and Gábor P. Nagy
Journal: Proc. Amer. Math. Soc. 142 (2014), 3029-3037
MSC (2010): Primary 20N05; Secondary 17B99, 20B15
DOI: https://doi.org/10.1090/S0002-9939-2014-12053-3
Published electronically: May 29, 2014
MathSciNet review: 3223359
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Abstract: We prove that every finite, commutative automorphic loop is solvable. We also prove that every finite, automorphic $ 2$-loop is solvable. The main idea of the proof is to associate a simple Lie algebra of characteristic $ 2$ to a hypothetical finite simple commutative automorphic loop. The ``crust of a thin sandwich'' theorem of Zel'manov and Kostrikin leads to a contradiction.


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Additional Information

Alexander Grishkov
Affiliation: Departamento de Matemática, Universidade de São Paulo, Caixa Postal 66281, São Paulo-SP, 05311-970, Brazil
Email: grishkov@ime.usp.br

Michael Kinyon
Affiliation: Department of Mathematics, University of Denver, 2360 S. Gaylord Street, Denver, Colorado 80208
Email: mkinyon@math.du.edu

Gábor P. Nagy
Affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
Email: nagyg@math.u-szeged.hu

DOI: https://doi.org/10.1090/S0002-9939-2014-12053-3
Keywords: Automorphic loops, Lie algebras of characteristic $2$, primitive groups
Received by editor(s): November 30, 2011
Received by editor(s) in revised form: September 27, 2012, and October 3, 2012
Published electronically: May 29, 2014
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2014 American Mathematical Society

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