Loops with exponent three in all isotopes

Details Loops with exponent three in all isotopes Loops with exponent three in all isotopes

2011-02-28

arxiv.org/abs/1103.0054v1

Mathematics

Group Theory

Scientific paper

It was shown by van Rees \cite{vR} that a latin square of order $n$ cannot have more than $n^2(n-1)/18$ 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, (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.

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