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.
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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.