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 $\operatorname{Stab} \left({x}\right)$ 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:
 * $X^g := \left\{ {x \in X: g x = x}\right\}$

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

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.