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Quasigroup automorphisms and the Norton-Stein complex
Author(s):
Brent
L.
Kerby;
Jonathan D. H.
Smith
Journal:
Proc. Amer. Math. Soc.
138
(2010),
3079-3088.
MSC (2010):
Primary 20N05, 05B15
Posted:
April 29, 2010
MathSciNet review:
2653932
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Abstract:
Suppose that  is the largest power of two that divides the order of a finite quasigroup  . It then follows that each automorphism of  must contain a cycle of length not divisible by  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.
References:
-
- 1.
- D. Bryant, M. Buchanan and I.M. Wanless, The spectrum for quasigroups with cyclic automorphisms and additional symmetries, Discrete Math. 309 (2009), 821-833. MR 2502191 (2010b:20122)
- 2.
- R.M. Falcón, Cycle structures of autotopisms of the Latin squares of order up to
 , to appear in Ars Combinatoria. arXiv:0709.2973v2 [math.CO], 2009. - 3.
- B. Fischer, Distributive Quasigruppen endlicher Ordnung, Math. Z. 83 (1964), 267-303. MR 0160845 (28:4055)
- 4.
- J.R. Munkres, Topology, Prentice-Hall, Englewood Cliffs, NJ, 2000.
- 5.
- J. Nielsen, Die Struktur periodischer Transformationen von Flächen, Kgl. Danske Videnskabernes Selskab., Math.-fys. Meddelelser 15 (1937), 1-77, translated as The structure of periodic surface transformations, pp. 65-102 in ``Jakob Nielsen: Collected Mathematical Papers, Vol. 2,'' Birkhäuser, Boston, MA, 1986. MR 0865336 (88a:01070b)
- 6.
- D.A. Norton and S.K. Stein, An integer associated with Latin squares, Proc. Amer. Math. Soc. 7 (1956), 331-334. MR 0077486 (17:1043f)
- 7.
- S.K. Stein, Homogeneous quasigroups, Pac. J. Math. 14 (1964), 1091-1102. MR 0170972 (30:1206)
- 8.
- S.K. Stein, On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957), 228-256. MR 0094404 (20:922)
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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:
10.1090/S0002-9939-10-10473-0
PII:
S 0002-9939(10)10473-0
Keywords:
Latin square,
quasigroup,
surface,
monodromy
Received by editor(s):
November 11, 2009
Posted:
April 29, 2010
Communicated by:
Jonathan I. Hall
Copyright of article:
Copyright
2010,
American Mathematical Society
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