Alphabeta MathFree Products and Amalgamation — Examples
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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.
✓ 8 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
C₂ free-product C₂ is the infinite dihedral group, and the product of its generators has infinite order
Example
C₂ free-product C₃ has presentation with only the two factor relations and is infinite
Example
The canonical surjection from a free product to the direct product of its factors
False statement
FALSE: a free product of abelian groups is abelian
Example
Amalgamating infinite cyclic groups by multiplication by m and n gives the presentation with relation xᵐ=yⁿ
Example
Amalgamating C₂ inside C₄ and C₆ gives the presentation with a²=b³
False statement
FALSE: canonical factor maps into every group pushout are injective
Example