Local Structure

Abstract

In many branches of mathematics, it is profitable to study an issue by somehow “localizing” with respect to a given prime number. In this chapter, we adapt this doctrine to group theory by studying finite groups through their subgroups of prime-power order. This notion of looking at the “local structure” of finite groups has proven to be very powerful. We start in Section 7 with Sylow’s theorem on subgroups of maximal prime-power order. Section 8 concentrates on the properties of finite groups of prime-power order. Section 9 gives an important application, the Schur-Zassenhaus theorem.