The order of automorphisms of quasigroups

Brendan Damien McKay, Ian Murray Wanless, Xiande Zhang

Research output: Contribution to journalArticleResearchpeer-review

8 Citations (Scopus)

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 Sn 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 Kn. 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.

Original languageEnglish
Pages (from-to)275-288
Number of pages14
JournalJournal of Combinatorial Designs
Volume23
Issue number7
DOIs
Publication statusPublished - 1 Jul 2015

Keywords

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

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