p-Groups and Sylow Theory

Abstract

The basic theory of p-groups (where all elements have order a power of p) is discussed, and the five Sylow theorems are derived—these results form one of the most important aspects of the finite theory. There are then two sections of applications, the first gives (a) some facts about groups whose orders have a small number of factors, (b) proves the so-called Frattini Argument, and (c) introduces nilpotent groups. The second is Web Section 6.5 which gives some more substantial applications including a proof of Burnside’s Normal Complement Theorem and a discussion of groups all of whose Sylow subgroups are cyclic.