TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29‡ rests on unproved material

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The word problem for finitely generated abelian groups is solvable

Statement

Every finitely generated abelian group has solvable word problem.

Facts & Assumptions

Given: A finitely generated abelian group $A$ with a fixed finite presentation.

‡ [L1]

Every finitely generated abelian group admits an invariant-factor decomposition $Z^{r} \times Z / d_{1} Z \times \dots \times Z / d_{k} Z$ with $d_{1} ∣ \dots ∣ d_{k}$ . (Finitely generated abelian groups admit invariant-factor normal form ‡)

Proof

technique · direct

1.1

By [L1], identify $A$ with $Z^{r} \times Z / d_{1} Z \times \dots \times Z / d_{k} Z$ . Any input word on a finite generating set evaluates, using commutativity, to one integer exponent sum in each of these finitely many coordinates.

‡ L1given

2.1

The word represents the identity exactly when the $Z^{r}$ coordinates are all $0$ and each torsion coordinate is congruent to $0$ modulo its invariant factor $d_{i}$ . Those are finitely many integer checks, so they give a terminating decision procedure.

step 1.1algebra

3.1

Therefore finitely generated abelian groups have solvable word problem.

step 2.1∎

Depends on

Used by

The word problem in a finite cyclic presentation Example

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jack Jeffries, Math 817: Introduction to Modern Algebra I (Fall 2025) (standard reference, not scraped)