  Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on
$\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that
$G$ \emph{synchronizes} $f$ if the transformation semigroup $\langle
G,f\rangle$ contains a constant map, and that $G$ is a \emph{synchronizing
group} if $G$ synchronizes \emph{every} non-permutation.
  A synchronizing group is necessarily primitive, but there are primitive
groups that are not synchronizing. Every non-synchronizing primitive group
fails to synchronize at least one uniform transformation (that is,
transformation whose kernel has parts of equal size), and it has previously
been conjectured that a primitive group synchronizes every non-uniform
transformation.
  The first goal of this paper is to prove that this conjecture is false, by
exhibiting primitive groups that fail to synchronize specific non-uniform
transformations of ranks $5$ and $6$. In addition we produce graphs whose
automorphism groups have approximately $\sqrt{n}$ \emph{non-synchronizing
ranks}, thus refuting another conjecture on the number of non-synchronizing
ranks of a primitive group.
  The second goal of this paper is to extend the spectrum of ranks for which it
is known that primitive groups synchronize every non-uniform transformation of
that rank. It has previously been shown that a primitive group of degree $n$
synchronizes every non-uniform transformation of rank $n-1$ and $n-2$, and here
this is extended to $n-3$ and $n-4$.
  Determining the exact spectrum of ranks for which there exist non-uniform
transformations not synchronized by some primitive group is just one of several
natural, but possibly difficult, problems on automata, primitive groups, graphs
and computational algebra arising from this work; these are outlined in the
final section.
