Loops with exponent three in all isotopes
It was shown by van Rees [Subsquares and transversals in latin squares, Ars Combin. 29B (1990) 193–204] that a latin square of order [Formula: see text] has at most [Formula: see text] latin subsquares of order [Formula: see text]. He conjectured that this bound is only achieved if [Formula: see text] is a power of [Formula: see text]. We show that it can only be achieved if [Formula: see text]. 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 [Formula: see text]. 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 [Formula: see text] 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.