{ "artifactId": "artifact.library.mathlib.cauchy-theorem-finite-groups.v001", "candidateOnly": false, "category": "Algebra", "claimBoundary": "This page indexes Mathlib's Fintype-cardinality form of Cauchy's theorem for finite groups: for [Group G], [Fintype G], prime p supplied by Fact, and p dividing Fintype.card G, there exists x : G with orderOf x = p. It asserts one existential witness; it does not assert uniqueness, classify the generated subgroup or any other subgroups, or state the nearby apostrophe variant using [Finite G] and Nat.card G.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem exists_prime_orderOf_dvd_card {G : Type*} [Group G] [Fintype G] (p : ℕ) [hp : Fact p.Prime] (hdvd : p ∣ Fintype.card G) : ∃ x : G, orderOf x = p", "family": "Finite group theory", "id": "library.mathlib.cauchy-theorem-finite-groups.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/cauchy-theorem-finite-groups/", "entryData": "/data/library-theorems/cauchy-theorem-finite-groups.json", "evidence": "/proofs/artifact.library.mathlib.cauchy-theorem-finite-groups.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.cauchy-theorem-finite-groups.v001.evidence.json", "source": "/sources/upstream/cauchy-theorem-finite-groups/" }, "landmark": { "completeFormalType": "theorem exists_prime_orderOf_dvd_card {G : Type*} [Group G] [Fintype G] (p : ℕ) [hp : Fact p.Prime] (hdvd : p ∣ Fintype.card G) : ∃ x : G, orderOf x = p", "concepts": [ "finite groups", "prime divisors", "element order", "product-one tuples", "cyclic rotation", "fixed-point counting" ], "plainLanguage": "If a prime p divides the number of elements in a finite group G, then G contains at least one element whose order is exactly p.", "poster": { "alt": "An ivory finite-group poster states the prime-divisibility hypothesis and exact element-order conclusion above product-one tuple bands, a cyclic-rotation chamber, a distinct constant witness tuple, and a five-step representative element orbit.", "byteSize": 2881921, "caption": "Every prime divisor of a finite group's cardinality occurs as the exact order of at least one element.", "description": "The poster foregrounds the Fintype.card divisibility implication and then follows the checked source route through length-p product-one tuples, cyclic rotation, prime fixed-point counting, a second constant fixed tuple, and extraction of an element of order p. Its footer keeps existence and the selected Fintype endpoint explicit.", "derivatives": [ { "byteSize": 133258, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cauchy-theorem-finite-groups/theorem-poster-v1-640.webp", "sha256": "sha256:4262ba5bb76ba4f754b663fb7c8d9133fccafd50e0c9d55b72bf1b1767f26228", "width": 640 }, { "byteSize": 295856, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cauchy-theorem-finite-groups/theorem-poster-v1-1024.webp", "sha256": "sha256:e572b3f0c1385b3038042ff38611356d9e0df58ce237e2adc7340dc1c65f411e", "width": 1024 } ], "generatedAt": "2026-07-25T16:37:57-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/cauchy-theorem-finite-groups/theorem-poster-v1.png", "sha256": "sha256:4c0a80ba500e1672753e4049fe8ca0ad839a3f63672d8a11340555ea0302212d", "title": "Cauchy's theorem for finite groups at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "ALGEBRA · FINITE GROUPS\nCAUCHY'S THEOREM\nG a finite group · p prime · p ∣ Fintype.card G\nTHERE EXISTS x ∈ G WITH orderOf x = p\nPRIME DIVISORS APPEAR AS ELEMENT ORDERS\nHOW THE CHECKED ROUTE MOVES\n1 · Form length-p tuples whose product is 1\n2 · Rotate the tuples cyclically\n3 · Use prime fixed-point counting to find a second fixed tuple\n4 · Read its constant entry as an element of order p\nEXACT SCOPE\nExistence only. No uniqueness or subgroup classification. Selected endpoint uses Fintype.card G." }, "visual": { "alt": "Many five-bead product-one tuple bands enter a gold rotation chamber, where five nonconstant patterned bands circulate around separate all-ivory and all-emerald fixed constant tuples before an emerald witness closes a five-step orbit.", "byteSize": 2853792, "caption": "Cyclic rotation of product-one p-tuples and prime fixed-point counting isolate a nonidentity constant tuple, whose entry has order p.", "description": "A dark engraved field sends varied product-one tuple candidates into a central cyclic action. Five patterned nonconstant bands circulate around two distinct constant fixed tuples; the emerald fixed tuple then condenses into one witness element with a closed representative prime-order orbit.", "derivatives": [ { "byteSize": 47404, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cauchy-theorem-finite-groups/theorem-schematic-v1-640.webp", "sha256": "sha256:8f0187d483a9e156db38a455a12525489215e48c5a1fc53e7086235d67c58e9b", "width": 640 }, { "byteSize": 160516, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cauchy-theorem-finite-groups/theorem-schematic-v1-1200.webp", "sha256": "sha256:287d8d921eaac0648f61d5e57df4cf51530afc3ac395abc7992c47ca16496c59", "width": 1200 } ], "generatedAt": "2026-07-25T16:37:57-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/cauchy-theorem-finite-groups/theorem-schematic-v1.png", "sha256": "sha256:8519238ccb04f0f2859a04128671497bf3f3286a9cedfcc1bea60e5ab075648e", "title": "Product-one tuples under prime cyclic rotation", "width": 1536 }, "walkthrough": null, "whyLandmark": "Cauchy's theorem reveals how the prime divisors of a finite group's size must appear inside the group's element structure. Mathlib's proof is especially instructive: a cyclic action on product-one tuples converts divisibility into a fixed point and then into an element of exact prime order." }, "nonClaims": [ "The selected declaration gives existence of an element of exact order p; it does not claim that the element is unique or canonical.", "The declaration does not classify the cyclic subgroup generated by the witness or any other subgroups of G.", "The selected endpoint uses [Fintype G] and Fintype.card G, not the nearby apostrophe variant using [Finite G] and Nat.card G.", "ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.cauchy-theorem-finite-groups.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": "A prime dividing the cardinality of a finite group is the exact order of some group element.", "targetId": "target.library.mathlib.cauchy-theorem-finite-groups.v001", "title": "Cauchy's Theorem for Finite Groups", "upstreamOrigin": { "declarationName": "exists_prime_orderOf_dvd_card", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/GroupTheory/Perm/Cycle/Type.lean#L496", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:6c1d87613176582bdee5b05a839be5de8f39a9b95e7d4cc349347475be226e13", "sourceByteLength": 34729, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "sourceLine": 496, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }