Loops with exponent three in all isotopes

Michael K Kinyon, Ian Murray Wanless

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It was shown by van Rees [Subsquares and transversals in latin squares, Ars Combin. 29B (1990) 193–204] that a latin square of order nn has at most n2(n−1)/18n2(n−1)/18 latin subsquares of order 33. He conjectured that this bound is only achieved if nn is a power of 33. We show that it can only be achieved if n≡3mod6n≡3mod6. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent 33. We call such loops van Rees loops and show that they form an equationally defined variety. We also show that: (1) In a van Rees loop, any subloop of index 3 is normal. (2) There are exactly six nonassociative van Rees loops of order 2727 with a nontrivial nucleus and at least 1 with all nuclei trivial. (3) Every commutative van Rees loop has the weak inverse property. (4) For each van Rees loop there is an associated family of Steiner quasigroups.
Original languageEnglish
Pages (from-to)1159-1177
Number of pages19
JournalInternational Journal of Algebra and Computation
Issue number7
Publication statusPublished - 2015


  • van Rees loop
  • quasigroup
  • latin square

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