Alphabeta MathHow 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.
Free abelian group on a set
Definition
A free abelian group on a set is an abelian group together with a map such that, for every abelian group and every function , there is a unique group homomorphism satisfying
Depends on
Used by
- The Cayley graph of ℤ for the generating set {1} is a line and its word metric is |m-n| Example
- The Cayley graph of ℤⁿ for the standard basis is the integer lattice, and its word metric is the sum of coordinate differences Example
- The free-abelian-group monad sends a set to its finite formal integer combinations Example
- The subgroup 2ℤ×ℤ has index two in ℤ² and its inclusion is a quasi-isometry Example
- The first potentially nonzero homotopy group of a wedge of higher spheres has its cell basis Lemma
- Aut(ℤⁿ)≅ GLₙ(ℤ) for every finite rank n Theorem
- The abelianisation of a free group on X is a free abelian group on X Theorem
Dependency tree · two levels
7 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
- John McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)