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 $\Stab x$ 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:
 * $X^g := \set {x \in X: g x = x}$

Then:
 * $\displaystyle \size {X / G} = \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}$

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

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