# Burnside's Lemma

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## 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.

## Proof

\(\ds \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}\) | \(=\) | \(\ds \frac 1 {\order G} \sum_{g \mathop \in G} \size {\set {x \in X: g x = x} }\) | by definition | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 {\order G} \sum_{x \mathop \in X} \size {\set {g \in G: g x = x} }\) | Same summation, different indexing | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 {\order G} \sum_{x \mathop \in X} \order {\Stab x}\) | Definition of Stabilizer | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 {\order G} \sum_{x \mathop \in X} \frac {\order G} {\order {\Orb x} }\) | Orbit-Stabilizer Theorem | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{x \mathop \in X} \frac 1 {\order {\Orb x} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{\Orb x \mathop \in X / G} \paren {\sum_{x \mathop \in \Orb x} \frac 1 {\order {\Orb x} } }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{\Orb x \mathop \in X / G} 1\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \order {X / G}\) |

$\blacksquare$

## Also known as

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

## Source of Name

This entry was named for William Burnside.