{ "artifactId": "artifact.library.mathlib.burnside-lemma.v001", "candidateOnly": false, "category": "Algebra", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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", "family": "Finite group actions", "id": "library.mathlib.burnside-lemma.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/burnside-lemma/", "entryData": "/data/library-theorems/burnside-lemma.json", "evidence": "/proofs/artifact.library.mathlib.burnside-lemma.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.burnside-lemma.v001.evidence.json", "source": "/sources/upstream/burnside-lemma/" }, "landmark": { "completeFormalType": "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", "concepts": [ "finite group actions", "fixed points", "orbit quotients", "stabilizers", "double counting", "equivalences of finite types" ], "plainLanguage": "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|.", "poster": { "alt": "An ivory and dark-green poster states the three finiteness hypotheses, the exact fixed-point cardinality identity, and the equivalence between fixed incidences and orbit–group pairs.", "byteSize": 2779289, "caption": "The sum of fixed-set cardinalities equals the number of orbits multiplied by the group cardinality.", "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 explicitly preserves the absence of a blanket finiteness assumption on β and of a division identity.", "derivatives": [ { "byteSize": 122772, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/burnside-lemma/theorem-poster-v1-640.webp", "sha256": "sha256:b538ca85ec877ddb6707ce6cb4d00cd8441a3ee470fcd786ccf2498cdbc6d6ec", "width": 640 }, { "byteSize": 269776, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/burnside-lemma/theorem-poster-v1-1024.webp", "sha256": "sha256:c1434752da007721ae5adfd4aec76b2be8b6472424b0177ec83f98d4006f4b93", "width": 1024 } ], "generatedAt": "2026-07-25T16:31:24-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/burnside-lemma/theorem-poster-v1.png", "sha256": "sha256:4c65431f6d05616cf10cc5906438e67429853d8217610a2f7fda18dfee2c6ed4", "title": "Burnside's Lemma at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "ALGEBRA · GROUP ACTIONS\nBURNSIDE'S LEMMA\nα acts on β\nα finite · every Fix(a) finite · Ω finite\nΣ a, |Fix(a)| = |Ω| × |α|\nCOUNT THE SAME PAIRS TWO WAYS\nΣ a, Fix(a) ≃ Ω × α\nΩ = orbit quotient\nEXACT SCOPE\nβ need not be finite. No division identity is asserted." }, "visual": { "alt": "A text-free engraved counting bridge carries varying fixed-point incidences for several group actions to orbit islands, each paired with an identical group rosette.", "byteSize": 3252661, "caption": "Fixed-point incidences and orbit–group pairs are two finite presentations of the same counted type.", "description": "On the left, separate action medallions govern point constellations with varying emerald fixed-point sets. Only the fixed incidences enter a central braided equivalence. On the right, the braid resolves into orbit-class islands, each carrying one congruent copy of the acting-group rosette, representing the product Ω × α without assuming all of β is finite.", "derivatives": [ { "byteSize": 58514, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/burnside-lemma/theorem-schematic-v1-640.webp", "sha256": "sha256:4dba78a6a8dfc4a2c9a1fd690a57857c502add2bbc9cc1c77601acc10746e4b0", "width": 640 }, { "byteSize": 205314, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/burnside-lemma/theorem-schematic-v1-1200.webp", "sha256": "sha256:2bee3845120c408afe2e2913da9acc80a541077d85fac75fdaee62c87a3f37c3", "width": 1200 } ], "generatedAt": "2026-07-25T16:31:24-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/burnside-lemma/theorem-schematic-v1.png", "sha256": "sha256:f8fe1649ba3b1dbc77960b74ee43b2d2868249284bfa3d64be27452c6cca3ff2", "title": "Burnside's Lemma schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "The selected declaration does not require or provide a blanket Fintype β instance.", "The selected declaration states a multiplication identity, not an average or division identity.", "The equivalence used by the proof is noncomputable and does not select a canonical orbit representative.", "Proof Atlas did not originate Burnside's lemma or Mathlib's declaration.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.burnside-lemma.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "For 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.", "targetId": "target.library.mathlib.burnside-lemma.v001", "title": "Burnside's Lemma", "upstreamOrigin": { "declarationName": "MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/GroupTheory/GroupAction/Quotient.lean#L262", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:50fc92cfeb4c8df97539ecbe4e6153518bea82afe6da61d531f92a9c0170ebb0", "sourceByteLength": 22247, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/GroupTheory/GroupAction/Quotient.lean", "sourceLine": 262, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }