Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 GG is an isomorphism G≅Cq0×⋯×Cqr−1,G\cong C_{q_0}\times\cdots\times C_{q_{r-1}}, where every qi>1q_i>1 is a prime power. The unordered multiset of the qiq_i, counted with multiplicity, is the elementary-divisor data. The cyclic factors and product use Every cyclic group is isomorphic to (Z,+)(\mathbb Z,+) or to (Z/n,+)(\mathbb Z/n,+) for its finite order n≥1n\ge1 and The external direct product G×HG\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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources