Talk 1: Between primitive and 2-transitive, I: synchronization and related properties

A permutation group $G$ on a finite set $\Omega$ is \emph{synchronizing} if, whenever $f:\Omega\to\Omega$ is not a permutation, the monoid generated by $G$ and $f$ contains a constant function (an element of rank~$1$). It is known that synchronizing groups are primitive, and indeed are ``basic'' in the O'Nan--Scott classification (that is, they preserve no cartesian power structure on $\Omega$). However, not all basic groups are synchronizing, and there are several variants of synchronization which define classes of groups between primitive and $2$-transitive. I will talk about these. I also hope to say a few words about the set of \emph{non-synchronizing ranks} of a permutation group: this set is empty if and only if the group is synchronizing, and is conjectured to be very small if the group is primitive (size at most $O(\log n)$, maybe), but large (at least $cn$) if $G$ is imprimitive. http://caul.cii.fc.ul.pt/slides/PeterCameron_talk1.pdf/