ABSTRACT
We will now apply the theory from the last chapter to obtain the Sylow theorems. These theorems will enable us to understand the p-subgroups of any finite group, taken one prime p at a time. Throughout this chapter we will assume that G is a finite group, and that it has pnm elements, where p is a prime, and m is relatively prime to p. We will be able to show that G has subgroups of order pn (called Sylow p-subgroups), that all such subgroups are isomorphic, and we will often be able to count how many of these subgroups there are.
Given our finite group G with pnm elements, suppose that H is a subgroup of G and a p-group. We then say that H is a p-subgroup of G. We call H a Sylow p-subgroup of G if it is a maximal p-subgroup of G. That is, if H is a Sylow p-subgroup of G and K is a p-subgroup of G that contains H, then K = H. Because G is a finite group, it is then obvious that every p-subgroup is contained in a Sylow p-subgroup. What is not yet obvious is how many elements a Sylow p-subgroup has.
