Entertainments

Abstract

We will now take the basic theory developed in the first part of this book, and play with it to show a little of what can be done with our theoretical machinery. This chapter is crudely divided into two parts concerning finite and infinite groups. This division is easy to make, but in fact is not really defensible. Some of the techniques for studying infinite groups consist of viewing them as limiting cases (or generalizations) of finite groups. On the other hand some geometric perspectives on group theory work just as well for finite as for infinite groups. Certain properties of finite groups have the aroma of their infinite relatives, whereas some of the strictly combinatorial (counting) arguments which flow from Sylow’s theorem have a pungent finite odour.