Having defined orbits and stabilizers, we now connect them by a precise counting principle. The orbit tells us how many points are reachable from $a$, while the stabilizer tells us how many group elements keep $a$ fixed. The orbit-stabilizer formula and orbit counting show that these two numbers are not independent: a large stabilizer forces a small orbit, and a large orbit forces a small stabilizer. This idea turns group actions into a powerful counting method for finite groups and finite sets. In this lesson we shall prove the orbit-stabilizer formula, derive the orbit decomposition formula, and use them to detect fixed points.

Let $G$ be a group, let $S$ be a $G$-set, and let $a\in S$. The \textbf{orbit size} of $a$ is the cardinality of $$\begin{equation*} [a]=\{g\cdot a:g\in G\}. \end{equation*}$$ The \textbf{stabilizer index} of $a$ is the index $$\begin{equation*} [G:G_a], \end{equation*}$$ where $G_a=\{g\in G:g\cdot a=a\}$ is the stabilizer of $a$.