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.
Example
The Klein four-group has the presentation
where and correspond to and .
Facts & Assumptions
Given: The presentation and the direct-product group .
A map of generators that sends every relator to the identity extends uniquely to a homomorphism from the presented group (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).
Every residue class modulo has exactly one representative in (For , every class in has one representative with , so ; while is in bijection with ).
Verification
Sending to and to kills the two square relators and the commutator in the abelian direct product, so [L1] constructs a homomorphism .
The commutator relation gives , and the square relations reduce both exponents modulo , so every element of has one of the forms .
Their images are , which are distinct and exhaustive by [L2]; together with step 1.2, this makes bijective.
Hence is isomorphic to , the Klein four-group.
Depends on
- Group presentation by generators and relations
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group
- For every natural $n$, $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold
- For $n\ge 1$, every class in $\mathbb{Z}/n$ has one representative $r$ with $0\le r<n$, so $\lvert\mathbb{Z}/n\rvert=n$; while $\mathbb{Z}/0$ is in bijection with $\mathbb{Z}$
- $G\times H$ is a group with identity $(e_G,e_H)$, coordinatewise inverses, and homomorphic coordinate projections
- Group and abelian group
- The principle of mathematical induction
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
32 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
- P. J. Cameron, Group Theory revision notes (standard reference, not scraped)
- J. Aspnes, Group Theory (standard reference, not scraped)