Latin squares and the Hall-Paige conjecture
The Hall-Paige conjecture deals with conditions under which a finite
group G will possess a complete mapping, or equivalently, a Latin
square based on the Cayley table of G will possess a
transversal. Two necessary conditions are known to be (i) that the
Sylow 2-subgroups of G are trivial or non-cyclic and (ii) that there
is some ordering of the elements of G which yields a trivial
product. These two conditions are known to be equivalent but
the first direct, elementary proof that (i) implies (ii) is given here.
It is also shown that the Hall-Paige conjecture implies the existence of a
duplex in every group table, thereby proving a special case of
Rodney's conjecture that every Latin square contains a duplex. A
duplex is a "double transversal", that is, a set of 2n entries in
a Latin square of order n such that each row, column and symbol is
represented exactly twice.