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Quasigroup automorphisms and the Norton-Stein complex


Authors: Brent L. Kerby and Jonathan D. H. Smith
Journal: Proc. Amer. Math. Soc. 138 (2010), 3079-3088
MSC (2010): Primary 20N05, 05B15
DOI: https://doi.org/10.1090/S0002-9939-10-10473-0
Published electronically: April 29, 2010
MathSciNet review: 2653932
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Abstract: Suppose that $ d>1$ is the largest power of two that divides the order of a finite quasigroup $ Q$. It then follows that each automorphism of $ Q$ must contain a cycle of length not divisible by $ d$ in its disjoint cycle decomposition. The proof is obtained by considering the action induced by the automorphism on a certain orientable surface originally described in a more restricted context by Norton and Stein.


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Additional Information

Brent L. Kerby
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Address at time of publication: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email: bkerby@math.byu.edu, kerby@math.utah.edu

Jonathan D. H. Smith
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: jdhsmith@iastate.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10473-0
Keywords: Latin square, quasigroup, surface, monodromy
Received by editor(s): November 11, 2009
Published electronically: April 29, 2010
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2010 American Mathematical Society