The Sylow Theorems
In this chapter, we will prove the Sylow theorems. These are difficult results, but fundamental to our understanding of the structure of finite groups. In particular, we will show that if \(p^n\) is the largest power of a prime p dividing the order of a finite group G, then G has at least one subgroup of order \(p^n\). Furthermore, we will discover that any two such subgroups are conjugate to each other, and determine a restriction upon the number of such subgroups. We will then explore various applications of these theorems, and conclude the chapter by classifying all groups of order smaller than 16.