Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Algebra

Burnside's Lemma

Let a finite group α act on β, and let Ω be the set of orbits. If every individual fixed-point set and Ω are finite, then adding the number of fixed points over all group elements gives |Ω| × |α|.

Exact theorem

Exact Mathlib statement

theorem MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group {α : Type u} {β : Type v} [Group α] [MulAction α β] [Fintype α] [∀ a : α, Fintype <| MulAction.fixedBy β a] [Fintype (Quotient <| MulAction.orbitRel α β)] : (∑ a : α, Fintype.card (MulAction.fixedBy β a)) = Fintype.card (Quotient <| MulAction.orbitRel α β) * Fintype.card α

The theorem at a glance

Burnside's Lemma at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

The sum of fixed-set cardinalities equals the number of orbits multiplied by the group cardinality. Explanatory diagram.
Detailed visual description

The poster places the separate finiteness hypotheses above the exact multiplication identity, then uses an unlabeled engraved bridge from varying fixed-point clusters to orbit islands paired with identical group rosettes. Its footer notes that the identity fixes all of β, so the hypotheses imply β is finite, and that the selected declaration states no division identity.

Statement structure

Statement and scope

Statement map for Burnside's LemmaFor a finite group action with finite fixed-point subtypes and finite orbit quotient, the total number of fixed incidences equals the number of orbits times the group cardinality. Claim boundary: This target indexes Mathlib's multiplication-form Burnside lemma for a group α acting on β. It assumes Fintype α, a Fintype instance for each fixed-point subtype fixedBy β a, and a Fintype instance for the orbit quotient Ω; it concludes that the sum of the fixed-set cardinalities equals Fintype.card Ω multiplied by Fintype.card α. The pinned upstream declaration is MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingFor a finite groupaction with finitefixed-point subtypes andfinite orbit quotient,the total number offixed incidences equalsthe number of orbitstimes the groupcardinality.Claim boundaryThis target indexesMathlib'smultiplication-formBurnside lemma for agroup α acting on β. Itassumes Fintype α, aFintype instance foreach fixed-point subtypefixedBy β a, and aFintype instance for theorbit quotient Ω; itconcludes that the sumof the fixed-setcardinalities equalsFintype.card Ωmultiplied byFintype.card α.Pinned declarationmathlib ·MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_groupStatement map for Burnside's LemmaFor a finite group action with finite fixed-point subtypes and finite orbit quotient, the total number of fixed incidences equals the number of orbits times the group cardinality. Claim boundary: This target indexes Mathlib's multiplication-form Burnside lemma for a group α acting on β. It assumes Fintype α, a Fintype instance for each fixed-point subtype fixedBy β a, and a Fintype instance for the orbit quotient Ω; it concludes that the sum of the fixed-set cardinalities equals Fintype.card Ω multiplied by Fintype.card α. The pinned upstream declaration is MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingFor a finite groupaction with finitefixed-point subtypes andfinite orbit quotient,the total number offixed incidences equalsthe number of orbitstimes the groupcardinality.Claim boundaryThis target indexesMathlib'smultiplication-formBurnside lemma for agroup α acting on β. Itassumes Fintype α, aFintype instance foreach fixed-point subtypefixedBy β a, and aFintype instance for theorbit quotient Ω; itconcludes that the sumof the fixed-setcardinalities equalsFintype.card Ωmultiplied byFintype.card α.Pinned declarationmathlib ·MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A concrete C₂ action presents the same ten objects as fixed incidences and as five orbit classes paired with two group elements. Explanatory scientific diagram.
Detailed visual description

The scientific schematic uses the action of C₂ on six points by one transposition. The two action rows contain six and four fixed points, while the orbit side contains four singleton classes and one two-point class, each paired with the two group elements. Thus the local example is a realizable instance of the multiplication-form Burnside identity.

Why it matters

A mathematical landmark

Burnside's lemma is a central bridge between symmetry and enumeration. Mathlib's proof exposes the underlying two-way count as an equivalence between the sigma type of fixed-point incidences and the product of the orbit quotient with the acting group.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's multiplication-form Burnside lemma for a group α acting on β. It assumes Fintype α, a Fintype instance for each fixed-point subtype fixedBy β a, and a Fintype instance for the orbit quotient Ω; it concludes that the sum of the fixed-set cardinalities equals Fintype.card Ω multiplied by Fintype.card α.

Source and local evidence

Where the theorem comes from

Existing declaration
MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.burnside-lemma.v001
Source
Open the pinned upstream reference