Quasigroup automorphisms and the Norton-Stein complex

Kerby, Brent; Smith, Jonathan D. H.

doi:10.1090/S0002-9939-10-10473-0

Quasigroup automorphisms and the Norton-Stein complex
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by Brent L. Kerby and Jonathan D. H. Smith PDF

Proc. Amer. Math. Soc. 138 (2010), 3079-3088 Request permission

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.

References

Darryn Bryant, Melinda Buchanan, and Ian M. Wanless, The spectrum for quasigroups with cyclic automorphisms and additional symmetries, Discrete Math. 309 (2009), no. 4, 821–833. MR 2502191, DOI 10.1016/j.disc.2008.01.020
R.M. Falcón, Cycle structures of autotopisms of the Latin squares of order up to $11$, to appear in Ars Combinatoria. arXiv:0709.2973v2 [math.CO], 2009.
Bernd Fischer, Distributive Quasigruppen endlicher Ordnung, Math. Z. 83 (1964), 267–303 (German). MR 160845, DOI 10.1007/BF01111162
J.R. Munkres, Topology, Prentice-Hall, Englewood Cliffs, NJ, 2000.
Jakob Nielsen, Jakob Nielsen: collected mathematical papers. Vol. 2, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1986. Edited and with a preface by Vagn Lundsgaard Hansen. MR 865336
D. A. Norton and Sherman K. Stein, An integer associated with Latin squares, Proc. Amer. Math. Soc. 7 (1956), 331–334. MR 77486, DOI 10.1090/S0002-9939-1956-0077486-1
Sherman K. Stein, Homogeneous quasigroups, Pacific J. Math. 14 (1964), 1091–1102. MR 170972
Sherman K. Stein, On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957), 228–256. MR 94404, DOI 10.1090/S0002-9947-1957-0094404-6