Latin Squares and the Hall-Paige Conjecture

The Hall–Paige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa 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 impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a ‘double transversal’,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60.