DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

Elementary-divisor data for a finite abelian group

Definition

An elementary-divisor decomposition of a finite abelian group $G$ is an isomorphism $G ≅ C_{q_{0}} \times \dots \times C_{q_{r - 1}},$ where every $q_{i} > 1$ is a prime power. The unordered multiset of the $q_{i}$ , counted with multiplicity, is the elementary-divisor data. The cyclic factors and product use Every cyclic group is isomorphic to $(Z, +)$ or to $(Z / n, +)$ for its finite order $n \geq 1$ and The external direct product $G \times H$ with componentwise multiplication. The data records factor isomorphism types, not distinguished internal subgroups; the trivial group has empty data.

Depends on

Used by

Dependency tree · two levels

11 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

Keith Conrad, Decomposition of Finite Abelian Groups, §§1-4 (standard reference, not scraped)
Richard Elman, Lectures on Abstract Algebra, Ch. 14 (standard reference, not scraped)