Abstract

Until 1980, there was no such subject as ‘infinite permutation groups’, according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10, 62], and a set of lecture notes by Wielandt [72], from the 1950s.

Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one.

I shall concentrate on a few topics, following the pattern of my conference lectures: the random graph (a case study); homogeneous relational structures (a powerful construction technique for interesting permutation groups); oligomorphic permutation groups (where the relations with other areas such as logic and combinatorics are clearest, and where a number of interesting enumerative questions arise); and the Urysohn space (another case study). I have preceded this with a short section introducing the language of permutation group theory, and I conclude with briefer accounts of a couple of topics that didn't make the cut for the lectures (maximal subgroups of the symmetric group, and Jordan groups).

I have highlighted a few specific open problems in the text. It will be clear that there are many wide areas needing investigation! I have also included some additional references not referred to in the text.

Notation and terminology

This section contains a few standard definitions concerning permutation groups. I write permutations on the right: that is, if g is a permutation of a set Ω, then the image of α under g is written αg.