After seeing that every group action produces a permutation representation, we now use actions for a deeper counting purpose. Instead of tracking an entire orbit directly, we count how many points are fixed by each group element and then average those numbers. This principle is Burnside's lemma, one of the most useful counting results in elementary group action theory. The focus keyword fixed points and Burnside's lemma connects two ideas: a fixed point records no movement under a group element, while Burnside's lemma converts fixed-point data into the number of orbits. In this lesson we shall define fixed points, prove Burnside's lemma, and use it to count orbits in finite actions.

Let $G$ be a group and let $S$ be a $G$-set. Let $g\in G$ and $a\in S$. Then $a$ is said to be \textbf{fixed by $g$} if $$\begin{equation*} g\cdot a=a. \end{equation*}$$ The set of all elements of $S$ fixed by $g$ is denoted by $$\begin{equation*} \operatorname{Fix}(g)=\{a\in S:g\cdot a=a\}. \end{equation*}$$