{
  "artifactId": "artifact.library.mathlib.first-sylow-theorem.v001",
  "candidateOnly": false,
  "category": "Algebra",
  "claimBoundary": "This target indexes Mathlib's subgroup-existence theorem for every prime-power divisor of a finite group's cardinality. It asserts one existential witness; it does not assert uniqueness, normality, conjugacy, a subgroup count, or the second and third Sylow theorems.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem Sylow.exists_subgroup_card_pow_prime {G : Type u} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime] (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n",
  "family": "Finite group theory",
  "id": "library.mathlib.first-sylow-theorem.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/first-sylow-theorem/",
    "entryData": "/data/library-theorems/first-sylow-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.first-sylow-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.first-sylow-theorem.v001.evidence.json",
    "source": "/sources/upstream/first-sylow-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem Sylow.exists_subgroup_card_pow_prime {G : Type u} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime] (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n",
    "concepts": [
      "finite groups",
      "subgroups",
      "prime powers",
      "group order",
      "normalizers"
    ],
    "plainLanguage": "Let p be prime. Whenever pⁿ divides the number of elements in a finite group G, there is at least one subgroup K with exactly pⁿ elements.",
    "poster": {
      "alt": "An ivory finite-group poster presents the prime-power divisibility hypothesis, one subgroup witness of the requested order, and the normalizer-quotient growth route.",
      "byteSize": 2915056,
      "caption": "Every prime power dividing a finite group order occurs as the exact order of a subgroup.",
      "description": "The poster centers the exact prime-power implication, then follows fixed cosets through a normalizer quotient to an order-p direction that lifts and enlarges a subgroup one rung at a time. Its footer keeps the existential boundary explicit.",
      "derivatives": [
        {
          "byteSize": 124040,
          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/first-sylow-theorem/theorem-poster-v1-640.webp",
          "sha256": "sha256:6c9c5b21a677f150f4189c6754d0e876cd775a1a1a7973bcf38bb517d3996760",
          "width": 640
        },
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          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/first-sylow-theorem/theorem-poster-v1-1024.webp",
          "sha256": "sha256:cb430d845d780f9bdcbafbf41b13e3cd7649676f69d6d2f274c6ce4989cfbb96",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-25T05:26:28.121Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/first-sylow-theorem/theorem-poster-v1.png",
      "sha256": "sha256:de77417684902b4cf1054f89ece66f75788a5498cdf1a53aa2b821bdf9ba54cf",
      "title": "First Sylow Theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "ALGEBRA · FINITE GROUPS\nFIRST SYLOW THEOREM\np prime · pⁿ divides |G|\nTHERE EXISTS K ≤ G WITH |K| = pⁿ\nEVERY DIVIDING PRIME POWER OCCURS\nHOW THE CHECKED ROUTE MOVES\n1 · Read fixed cosets through a normalizer quotient\n2 · Find an order-p direction in that quotient\n3 · Lift it to enlarge a subgroup by one prime factor\n4 · Climb from the trivial subgroup to order pⁿ\nEXACT SCOPE\nExistence only. No uniqueness, normality, conjugacy, or subgroup count."
    },
    "visual": {
      "alt": "A finite-group field contains one highlighted nested sequence of subgroup regions whose cardinalities grow by factors of a prime until reaching a requested prime-power divisor.",
      "byteSize": 3260359,
      "caption": "A prime-power divisor of a finite group order is realized by the cardinality of a subgroup.",
      "description": "A broad finite group is rendered as a field of discrete elements. One noncanonical witness route grows through nested subgroup boundaries by one factor of p at a time, ending at a highlighted subgroup of cardinality pⁿ. Several faint alternative subgroup contours prevent the witness from appearing unique.",
      "derivatives": [
        {
          "byteSize": 74584,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/first-sylow-theorem/theorem-schematic-v1-640.webp",
          "sha256": "sha256:3520c27dd304a49c91f38a16c0f620dc39472d5edbc4f84988e74764abe1e0d4",
          "width": 640
        },
        {
          "byteSize": 233126,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/first-sylow-theorem/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:ec1694b2568492bb130d52f4188131675d0b73f48dd12e1f7c2450f86ccc7859",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T05:26:28.121Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/first-sylow-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:782bfc68c1fbd0f9c947f9d1c63e85e0b886bb5189c75c7d3abaf2575a04d21a",
      "title": "First Sylow Theorem schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Sylow's theorems organize the prime-power structure hidden inside finite groups. Mathlib's selected declaration gives the strong existence form for every prime-power divisor, not only the largest power of p dividing the group order."
  },
  "nonClaims": [
    "Proof Atlas did not originate Sylow's theorem or Mathlib's declaration.",
    "The subgroup is not claimed to be unique, canonical, normal, or represented as a value of Mathlib's Sylow structure.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.first-sylow-theorem.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": "If a prime power divides the order of a finite group, the group has a subgroup of exactly that prime-power order.",
  "targetId": "target.library.mathlib.first-sylow-theorem.v001",
  "title": "First Sylow Theorem",
  "upstreamOrigin": {
    "declarationName": "Sylow.exists_subgroup_card_pow_prime",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/GroupTheory/Sylow.lean#L647",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:065f21a50a881e60f2353ebc1fb9b75b6afe88faeab891883af07bcae93754a6",
    "sourceByteLength": 40323,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/GroupTheory/Sylow.lean",
    "sourceLine": 647,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}