##### Theorem15.1Cauchy

Let \(G\) be a finite group and \(p\) a prime such that \(p\) divides the order of \(G\). Then \(G\) contains a subgroup of order \(p\).

We will use what we have learned about group actions to prove the Sylow Theorems. Recall for a moment what it means for \(G\) to act on itself by conjugation and how conjugacy classes are distributed in the group according to the class equation, discussed in Chapter 14. A group \(G\) acts on itself by conjugation via the map \((g,x) \mapsto gxg^{-1}\). Let \(x_1, \ldots, x_k\) be representatives from each of the distinct conjugacy classes of \(G\) that consist of more than one element. Then the class equation can be written as \begin{equation*}|G| = |Z(G)| + [G: C(x_1) ] + \cdots + [ G: C(x_k)],\end{equation*} where \(Z(G) = \{g \in G : gx = xg \text{ for all } x \in G\}\) is the center of \(G\) and \(C(x_i) = \{ g \in G : g x_i = x_i g \}\) is the centralizer subgroup of \(x_i\).

We begin our investigation of the Sylow Theorems by examining subgroups of order \(p\), where \(p\) is prime. A group \(G\) is a *\(p\)-group* if every element in \(G\) has as its order a power of \(p\), where \(p\) is a prime number. A subgroup of a group \(G\) is a *\(p\)-subgroup* if it is a \(p\)-group.

Let \(G\) be a finite group and \(p\) a prime such that \(p\) divides the order of \(G\). Then \(G\) contains a subgroup of order \(p\).

Let \(G\) be a finite group. Then \(G\) is a \(p\)-group if and only if \(|G| = p^n\).

Let us consider the group \(A_5\). We know that \(|A_5| = 60 = 2^2 \cdot 3 \cdot 5\). By Cauchy's Theorem, we are guaranteed that \(A_5\) has subgroups of orders \(2\), \(3\) and \(5\). The Sylow Theorems will give us even more information about the possible subgroups of \(A_5\).

We are now ready to state and prove the first of the Sylow Theorems. The proof is very similar to the proof of Cauchy's Theorem.

Let \(G\) be a finite group and \(p\) a prime such that \(p^r\) divides \(|G|\). Then \(G\) contains a subgroup of order \(p^r\).

A *Sylow \(p\)-subgroup* \(P\) of a group \(G\) is a maximal \(p\)-subgroup of \(G\). To prove the other two Sylow Theorems, we need to consider conjugate subgroups as opposed to conjugate elements in a group. For a group \(G\), let \({\mathcal S}\) be the collection of all subgroups of \(G\). For any subgroup \(H\), \({\mathcal S}\) is a \(H\)-set, where \(H\) acts on \({\mathcal S}\) by conjugation. That is, we have an action
\begin{equation*}H \times {\mathcal S} \rightarrow {\mathcal S}\end{equation*}
defined by
\begin{equation*}h \cdot K \mapsto hKh^{-1}\end{equation*}
for \(K\) in \({\mathcal S}\).

The set
\begin{equation*}N(H) = \{ g \in G : g H g^{-1} = H\}\end{equation*}
is a subgroup of \(G\) called the the *normalizer* of \(H\) in \(G\). Notice that \(H\) is a normal subgroup of \(N(H)\). In fact, \(N(H)\) is the largest subgroup of \(G\) in which \(H\) is normal.

Let \(P\) be a Sylow \(p\)-subgroup of a finite group \(G\) and let \(x\) have as its order a power of \(p\). If \(x^{-1} P x = P\), then \(x \in P\).

Let \(H\) and \(K\) be subgroups of \(G\). The number of distinct \(H\)-conjugates of \(K\) is \([H:N(K) \cap H]\).

Let \(G\) be a finite group and \(p\) a prime dividing \(|G|\). Then all Sylow \(p\)-subgroups of \(G\) are conjugate. That is, if \(P_1\) and \(P_2\) are two Sylow \(p\)-subgroups, there exists a \(g \in G\) such that \(g P_1 g^{-1} = P_2\).

Let \(G\) be a finite group and let \(p\) be a prime dividing the order of \(G\). Then the number of Sylow \(p\)-subgroups is congruent to \(1 \pmod{p}\) and divides \(|G|\).

Peter Ludvig Mejdell Sylow was born in 1832 in Christiania, Norway (now Oslo). After attending Christiania University, Sylow taught high school. In 1862 he obtained a temporary appointment at Christiania University. Even though his appointment was relatively brief, he influenced students such as Sophus Lie (1842–1899). Sylow had a chance at a permanent chair in 1869, but failed to obtain the appointment. In 1872, he published a 10-page paper presenting the theorems that now bear his name. Later Lie and Sylow collaborated on a new edition of Abel's works. In 1898, a chair at Christiania University was finally created for Sylow through the efforts of his student and colleague Lie. Sylow died in 1918.