The Structure of the Symmetric Groups

Abstract

The present chapter studies the symmetric groups with particular emphasis on the infinite case. Of course we know that every group is isomorphic to a subgroup of some symmetric group (for example, via its regular permutation representation), so one might suppose that it is not possible to say much useful about the symmetric groups unless we know a great deal about groups in general. However this is not true. There are certain facts which are available without a detailed knowledge of all of the subgroups, much in the same way that we have useful results about the set of real numbers without knowing detailed facts about individual real numbers.