Burnside's lemma

Burnside's lemma counts the orbits of a finite group action by averaging, over all group elements, the number of points each element fixes.

Group action: A rule assigning each element of a group a permutation of a set, compatibly with group multiplication. The lemma applies to points moved by a group action.
Pólya enumeration theorem: A theorem that counts colorings up to symmetry using the cycle structures of group elements. It extends Burnside's counting method to track how many times each color is used.
Finite group: A group containing finitely many elements, with an associative operation, identity, and inverses. The averaging formula sums over a finite set of symmetries.
William Burnside: An English mathematician known for major contributions to finite group theory. His textbook popularized the orbit-counting formula under his name.
Orbit: The set of points reachable from one point under the elements of a group action. Burnside's lemma counts these equivalence classes of points.
Necklace problem: The problem of counting bead colorings of a circular necklace up to rotation or other specified symmetries. Rotations identify many colorings, and Burnside's lemma counts the distinct necklaces.
Equivalence relation: A relation that is reflexive, symmetric, and transitive, partitioning a set into equivalence classes. Being in the same orbit is an equivalence relation on the acted-on set.
Ferdinand Georg Frobenius: A German mathematician whose work shaped group theory and representation theory. He stated an early version of the orbit-counting result before Burnside's textbook.
Stabilizer subgroup: The subgroup of group elements that leave a specified point unchanged. Fixed-point counts can be organized by which elements stabilize each point.
Bracelet problem: The problem of counting circular color patterns up to rotations and reflections. Adding reflections changes the acting group and its fixed-coloring counts.

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