Loops with exponent three in all isotopes

Abstract

It was shown by van Rees \cite{vR} that a latin square of order $n$ has at most $n^2(n-1)/18$ 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 $n\equiv3\bmod6$. 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 \emph{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 and at least 1 with all nuclei trivial, (3) Every commutative van Rees loop has the weak inverse property and (4) For each van Rees loop there is an associated family of Steiner quasigroups.