  The Hall--Paige conjecture asserts that a finite group has a complete mapping
if and only if its Sylow subgroups are not cyclic. The conjecture is now
proved, and one aim of this paper is to document the final step in the proof
(for the sporadic simple group $\mathrm{J}_4$).
  We apply this result to prove that primitive permutation groups of simple
diagonal type with three or more simple factors in the socle are
non-synchronizing. We also give the simpler proof that, for groups of affine
type, or simple diagonal type with two socle factors, synchronization and
separation are equivalent.
  Synchronization and separation are conditions on permutation groups which are
stronger than primitivity but weaker than $2$-homogeneity, the second of these
being stronger than the first. Empirically it has been found that groups which
are synchronizing but not separating are rather rare. It follows from our
results that such groups must be primitive of almost simple type.
