  J.M. Howie, the influential St Andrews semigroupist, claimed that we value an
area of pure mathematics to the extent that (a) it gives rise to arguments that
are deep and elegant, and (b) it has interesting interconnections with other
parts of pure mathematics.
  This paper surveys some recent results on the transformation semigroup
generated by a permutation group $G$ and a single non-permutation $a$. Our
particular concern is the influence that properties of $G$ (related to
homogeneity, transitivity and primitivity) have on the structure of the
semigroup. In the first part of the paper, we consider properties of $S=<G,a>$
such as regularity and idempotent generation. The second is a brief report on
the synchronization project, which aims to decide in what circumstances $S$
contains an element of rank 1. The paper closes with a list of open problems on
permutation groups and linear groups, and some comments about the impact on
semigroups are provided.
  These two research directions outlined above lead to very interesting and
challenging problems on primitive permutation groups whose solutions require
combining results from several different areas of mathematics, certainly
fulfilling both of Howie's elegance and value tests in a new and fascinating
way.
