Burnside's Lemma

Theorem
Let $G$ be a finite group acting on a set $X$.

Let $X/G$ be the set of orbits under this action.

For $x \in X$, let $\operatorname{Stab} (x)$ be the stabilizer of $x$ by $G$.

For $g \in G$, let $X^g$ denotes the set of all elements in $X$ which is fixed by $g$, that is:
 * $X^g := \left\{x \in X \mid gx = x \right\}$

Then:
 * $\displaystyle \left\vert X/G \right\vert = \frac{1}{\vert G \vert} \sum_{g \mathop \in G} \left\vert X^g \right\vert$

In words, number of orbits equals the average number of fixed elements.

Also known as
This theorem is also known as Burnside's Counting Theorem.