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.