Maximal Subgroups of Symmetric Groups

A theorem of O’Nan and Scott [6]; [2, Chapter 4] restricts the possibilities for maximal subgroups of finite symmetric groups: they are of six types of which the first four are explicitly known, the fifth involves a finite simple group, and the sixth an action of a simple group. This result, in conjunction with the Classification of Finite Simple Groups, has a number of consequences. In particular, it has been determined which subgroups on the list are actualy maximal [3]. In the infinite case, things are different: the O’Nan–Scott theorem does not hold in general, and the symmetric group has subgroups which lie in no maximal subgroup. However, versions of the O’Nan–Scott theorem are known for some classes of groups, and a number of results are known about maximal subgroups [4, 5]. Sometimes the precise form of the result depends on set-theoretic assumptions [1]. The project will involve summarising known results and working detailed examples.