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Keywords:

  • quasigroup;
  • automorphism;
  • Latin square;
  • autotopism;
  • autoparatopism;
  • Steiner triple system;
  • 1-factorization MSC Classification: 05B15;
  • 20N05 (05B05;
  • 05C70)

Abstract

We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1-factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group inline image will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1-factorization of the complete graph inline image. Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D.  Stones.