Introduction

In a lecture at Oberwolfach in the 1980s with the title “Topology in permutation groups”, Helmut Wielandt defended the proposition that topology in permutation groups is of no use, and came to the conclusion that (with the addition of the word “almost”) this was indeed the case. Wielandt made it clear that he was not speaking about the topology of permutation groups. (There is a natural topology on the symmetric group, namely the topology of pointwise convergence. In the case of countable degree, the open subgroups are those lying between the pointwise and setwise stabiliser of a finite set, and the closed subgroups are the automorphism groups of first-order structures.) Nevertheless, many in the audience felt that Wielandt's conclusion was too pessimistic. Since then, further results have supported this view. The present paper is a survey of some of these results. I begin with an account of Wielandt's own work on the subject, the connection of non-Hausdorff topologies preserved by G and the notions of primitivity and strong primitivity. The next topic is a theorem of Macpherson and Praeger, according to which a primitive group which preserves no non-trivial topology is highly transitive. Their proof uses some deep results from model theory. I give an elementary argument to replace part of the proof.

Some topologies low in the separation hierarchy can be interpreted as relational structures (specifically, preorders), so that their homeomorphism groups are closed subgroups of the symmetric group.