RemarkRemark: Literature-sourcedProof: Not suppliedSession-authored (Fable 5 assisted)audited 2026-08-29‡ sources checked 2026-08-29‡ not proved here

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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Finitely generated abelian groups admit invariant-factor normal form

Statement

Every finitely generated abelian group $A$ is isomorphic to

$Z^{r} \times Z / d_{1} Z \times \dots \times Z / d_{k} Z,$

where $r \geq 0$ , each $d_{i} > 1$ , and $d_{1} ∣ d_{2} ∣ \dots ∣ d_{k}$ .

Remarks

This page records only the exact normal form. It does not prove the full classification theorem.

The consequence used below is algorithmic: once a finitely generated abelian group has been written in these coordinates, equality with the identity reduces to checking an integer vector and finitely many residue classes.

Used by

The word problem for finitely generated abelian groups is solvable Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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