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 products with amalgamation along monomorphisms
Definition
If and are injective homomorphisms, their pushout is called the free product with amalgamation and is denoted . The quotient construction is A group pushout is the quotient of a free product by the amalgamating relations, and injectivity means the trivial-kernel condition of A group homomorphism is injective if and only if its kernel is trivial. The notation anticipates identifying with its two images, but injectivity of the canonical maps is a theorem, not part of this definition.
Depends on
Used by
- Amalgamation over the trivial group is the ordinary free product Corollary
- Transversal normal-form data for an amalgamated free product Definition
- Conventions and proved scope for free products and amalgamation Remark
- A free product with amalgamation has the factor presentations plus the amalgamating relations Theorem
- A one-segment graph of groups gives an amalgamated free product Theorem
Dependency tree · two levels
13 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
- George D. Torres, Combinatorial Group Theory, §2 (standard reference, not scraped)
- B. H. Neumann, Lectures on Topics in the Theory of Infinite Groups, Ch. 9 (standard reference, not scraped)