Loops with exponent three in all isotopes
It was shown by van Rees that a latin square of order n
cannot have more than n2(n-1)/18 latin subsquares of order 3. He
conjectured that this bound is only achieved if n is a power of
3. We show that it can only be achieved if n=3 mod 6. 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 3. 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 6 nonassociative van Rees loops of order
27 with a non-trivial nucleus, (3) There is a Steiner quasigroup
associated with every van Rees loop and (4) Every Bol loop of exponent
3 is a van Rees loop.