The Sylow Theorems

We have already observed (see statements after the proof of Theorem 6.2; also see Exercise 4 for Section 6.2) that the converse of Lagrange's theorem is false, i.e., if $G$ is a finite group of order $n$ and if $d\vert n$, then $G$ need not contain a subgroup of order $d$. If $d$ is a prime $p$ or a power of a prime $p^e$, however, then we shall see that $G$ must contain subgroups of that order. In particular, we shall see that if $p^d$ is the highest power of $p$ that divides $n$, than all subgroups of that order are actually conjugate, and we shall finally get a formula concerning the number of such subgroups. These theorems constitute the Sylow Theorems which, along with a few applications, will be the matter of concern of this chapter.