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Searching for small simple automorphic loops

Part of: Permutation groups Other generalizations of groups

Published online by Cambridge University Press:  01 August 2011

Kenneth W. Johnson ,
Michael K. Kinyon ,
Gábor P. Nagy  and
Petr Vojtěchovský
Kenneth W. Johnson
Affiliation:
Penn State Abington, 1600 Woodland Rd, Abington PA 19001, USA (email: kwj1@psu.edu)
Michael K. Kinyon
Affiliation:
Department of Mathematics, University of Denver, 2360 S Gaylord St, Denver Colorado 80112, USA (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)
Petr Vojtěchovský
Affiliation:
Department of Mathematics, University of Denver, 2360 S Gaylord St, Denver Colorado 80112, USA (email: petr@math.du.edu)

Abstract

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A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 212, and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-associative simple right automorphic loops. We also prove that every automorphic loop has the antiautomorphic inverse property, and that a right automorphic loop is automorphic if and only if its conjugations are automorphisms.

MSC classification

Secondary: 20N05: Loops, quasigroups 20B15: Primitive groups 20B40: Computational methods

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 200 - 213
Copyright
Copyright © London Mathematical Society 2011

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