ABSTRACT
In this chapter we present a powerful idea in group theory which is group action. The consequences of group action are many and the applications are incredibly useful as we shall see in the later sections in this chapter. Indeed, if someone wants to see concrete applications in group theory, then group action certainly fits the bill. In Section 4.1 we introduce the notion of group action. In Sections 4.2 and 4.3 we see a very nice application of group action in the field of combinatorics. Section 4.4 presents some deep theoretical consequences which follow from group action. The important Sylow Theorems in Section 4.5 follow from group action. We then use the Sylow Theorems to classify finite groups of particular sizes in Section 4.6, which in turn assists us in classifying completely finite abelian groups in Section 4.7.
