{ "artifactId": "artifact.library.mathlib.finitely-generated-abelian-groups.v001", "candidateOnly": false, "category": "Algebra", "claimBoundary": "This target indexes Mathlib's existence theorem: a finitely generated additive commutative group is additively equivalent to a finite-rank free integer part times a finite direct sum of cyclic prime-power groups. This declaration does not assert uniqueness, canonical factors, invariant-factor ordering, or an algorithm for computing the decomposition.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem AddCommGroup.equiv_free_prod_directSum_zmod (G : Type u) [AddCommGroup G] [hG : AddGroup.FG G] : ∃ (n : ℕ) (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime (p i)) (e : ι → ℕ), Nonempty (G ≃+ (Fin n →₀ ℤ) × ⨁ i : ι, ZMod (p i ^ e i))", "family": "Abelian group structure", "id": "library.mathlib.finitely-generated-abelian-groups.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/finitely-generated-abelian-groups/", "entryData": "/data/library-theorems/finitely-generated-abelian-groups.json", "evidence": "/proofs/artifact.library.mathlib.finitely-generated-abelian-groups.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.finitely-generated-abelian-groups.v001.evidence.json", "source": "/sources/upstream/finitely-generated-abelian-groups/" }, "landmark": { "completeFormalType": "theorem AddCommGroup.equiv_free_prod_directSum_zmod (G : Type u) [AddCommGroup G] [hG : AddGroup.FG G] : ∃ (n : ℕ) (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime (p i)) (e : ι → ℕ), Nonempty (G ≃+ (Fin n →₀ ℤ) × ⨁ i : ι, ZMod (p i ^ e i))", "concepts": [ "finitely generated abelian groups", "free abelian groups", "torsion", "cyclic prime-power groups", "direct sums", "additive equivalence" ], "plainLanguage": "Every finitely generated abelian group has the same additive structure as a finite-rank free integer group together with finitely many cyclic groups whose orders are prime powers.", "poster": { "alt": "An ivory algebra poster splits a finitely generated abelian group through a two-way equivalence into an integer lattice and a finite direct sum of cyclic prime-power rings.", "byteSize": 2790519, "caption": "Finite generation forces an additive equivalence with a finite-rank free integer part and finitely many cyclic prime-power components.", "description": "The exact existential decomposition identifies a natural number n, a finite index type, primes pᵢ, exponents eᵢ, and an additive equivalence. The checked route applies the finitely generated module structure theorem over the integers and converts cyclic quotient modules into ZMod prime-power factors.", "derivatives": [ { "byteSize": 134058, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/finitely-generated-abelian-groups/theorem-poster-v1-640.webp", "sha256": "sha256:ca20557c22049d7b3f59270788afee4cbf834e5a8e2dbbbbc15fff7872381819", "width": 640 }, { "byteSize": 286340, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/finitely-generated-abelian-groups/theorem-poster-v1-1024.webp", "sha256": "sha256:5b93311ff672b0d5fc2bc1eae7678c317ec05f5b09ba5cec0004add62d90593f", "width": 1024 } ], "generatedAt": "2026-07-25T05:49:45.723Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/finitely-generated-abelian-groups/theorem-poster-v1.png", "sha256": "sha256:fb1167900a33c36b7bc2912d4a315a984fd39bdae966b7ab9f563515ad72aef2", "title": "Finitely Generated Abelian Groups — free rank and prime-power torsion", "width": 1024, "kind": "theorem_poster", "transcript": "ALGEBRA · ABELIAN GROUP STRUCTURE\nSTRUCTURE THEOREM FOR FINITELY GENERATED ABELIAN GROUPS\nG is a finitely generated abelian group\nG ≃+ (Fin n →₀ ℤ) × ⨁ᵢ ZMod(pᵢ^{eᵢ})\nfor some n, finitely many indices i, prime pᵢ, and exponent eᵢ\nFREE PART\nA finite-rank integer lattice (Fin n →₀ ℤ)\nTORSION PART\nA finite direct sum of cyclic prime-power groups ZMod(pᵢ^{eᵢ})\nHOW THE CHECKED ROUTE MOVES\n1 · Regard G as a finitely generated module over ℤ\n2 · Apply the module structure theorem over a principal ideal domain\n3 · Separate the free summand from cyclic quotient summands\n4 · Replace each integer prime-power quotient by the corresponding ZMod group\n5 · Assemble an additive equivalence with G\nEXACT SCOPE\nExistence of an additive equivalence. No uniqueness, canonical ordering, distinct-prime requirement, or decomposition algorithm." }, "visual": { "alt": "One woven abelian group lattice passes through an equivalence gate and separates into straight integer-grid directions and a finite constellation of cyclic prime-power rings.", "byteSize": 2688188, "caption": "Every finitely generated abelian group is additively equivalent to a finite free integer part times finitely many cyclic prime-power parts.", "description": "A unified commutative group structure separates into an unbounded integer lattice and finitely many closed cyclic components, each organized around one prime-power order. The relationship is drawn as an invertible equivalence rather than as literal equality or a canonical disassembly.", "derivatives": [ { "byteSize": 58784, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/finitely-generated-abelian-groups/theorem-schematic-v1-640.webp", "sha256": "sha256:98008c3be80c5371a8a6630f34bd9518399697e18f6b2fe321e776262c43c91e", "width": 640 }, { "byteSize": 173274, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/finitely-generated-abelian-groups/theorem-schematic-v1-1200.webp", "sha256": "sha256:fd086f234a20fe613273dc37491c713463054a45ebfc11a8c1de954f817609b6", "width": 1200 } ], "generatedAt": "2026-07-25T06:20:33.299Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/finitely-generated-abelian-groups/theorem-schematic-v1.png", "sha256": "sha256:131d72f509f079ce70565507b392230b813e0fa4e5b52654f4adf8128b9c8229", "title": "A finitely generated abelian group separates into free and prime-power parts", "width": 1536 }, "walkthrough": null, "whyLandmark": "The structure theorem turns every finitely generated abelian group into explicit free and prime-power building blocks. It is a central classification result linking group theory, modules over principal ideal domains, and arithmetic." }, "nonClaims": [ "The selected declaration proves existence of a decomposition, not uniqueness or canonicity.", "It does not order, merge, or require distinct prime-power factors.", "The formal endpoint is an additive equivalence, not literal definitional equality of types.", "It provides no algorithm for extracting the factors from a presentation.", "ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.finitely-generated-abelian-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": "Every finitely generated abelian group decomposes into a finite-rank free part and finitely many cyclic prime-power parts.", "targetId": "target.library.mathlib.finitely-generated-abelian-groups.v001", "title": "Structure Theorem for Finitely Generated Abelian Groups", "upstreamOrigin": { "declarationName": "AddCommGroup.equiv_free_prod_directSum_zmod", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/GroupTheory/FiniteAbelian/Basic.lean#L119", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:02b8058bf2c84ec16e519370f60044def7ce5060432c30cd30fdb7849d45f648", "sourceByteLength": 11685, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/GroupTheory/FiniteAbelian/Basic.lean", "sourceLine": 119, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }