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Section15.1The Sylow Theorems

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.

Proof
Example15.3

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.

Proof

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.

Proof
Proof

SubsectionHistorical Note

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.