In these quizzes, we'll explore the theory of **group actions**.

Most groups act naturally on some set: \(D_n\) acts on an \(n\)-gon, \(S_n\) acts on \(\{1,2,\ldots,n\},\) \(GL_n({\mathbb R})\) acts on \({\mathbb R}^n\) (via left multiplication), and so on.

We'll apply this theory to **Burnside's Lemma**, **Semidirect products**, and the **Sylow theorems**.

**Burnside's lemma** is a useful way to count combinatorial objects "up to" some operation. For example, there are 16 squares that can be made with blue and yellow vertices, but if we consider two squares the same if one can be rotated to produce the other, the question of determining the number of distinct squares becomes less trivial.

Burnside's lemma gives an elegant and easy-to-calculate formula for this number, in a somewhat surprising way--instead of counting elements of the set, Burnside's lemma shows that the computation can be done via a sum running over elements of the group, looking at fixed points only.

**Semidirect products**, are generalizations of the direct product construction for groups. Semidirect products are extremely powerful tools for building larger groups from smaller ones, and for classifying groups based on properties of their subgroups.

The **Sylow theorems** are extremely effective tools for analyzing the structure of finite groups. They can be used to analyze and classify finite groups, by looking at the properties of their Sylow subgroups.

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