##  [Burnside's Lemma](/burnsides-lemma-2) 

 Definition

A counting lemma that computes the number of distinct orbits of a finite set under the action of a finite group by averaging the number of fixed points of group elements.

 

 

 

 

 

 





## Principle

Principle

The number of orbits equals the average, over group elements, of the size of their fixed-point sets: |X/G| = (1/|G|) ∑_{g∈G} |Fix(g)|.

 

 

 

 

 





## Demonstration

Demonstration

Count colorings of the vertices of a square up to rotational symmetries by averaging the number of colorings fixed by each rotation in the rotation group of the square.

 

 

 

 

## Misapplication

Misapplication

Averaging fixed points over a collection that is not a group, or over an infinite group without appropriate normalization, which yields meaningless or incorrect orbit counts.

 

 

 

 

 





## Consequence

Consequence

Provides an effective and often simple method to enumerate distinct configurations under symmetry, reducing orbit counting to fixed-point calculations for each group element.

 

 

 

 

## Reversal

Reversal

Viewing the lemma in reverse emphasizes characterizing each group element by its cycle structure; Pólya's theorem reverses or generalizes this by encoding these structures in the cycle index polynomial.

 

 

 

 

 





## Boundary

Boundary

Requires a finite group action on a finite set; not applicable if the action is not by permutations or if orbits are infinite without further structure.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with orbit–stabilizer counting or brute-force classification: Burnside reduces global orbit counting to local fixed-point checks, while orbit–stabilizer focuses on individual element stabilizers and orbit sizes.

 

 

 

 

 





## Synthesis

Synthesis

Burnside's Lemma unifies symmetry and counting by converting a global enumeration problem into an average of local invariances, forming the conceptual bridge to cycle-index methods like Pólya's theorem.