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.

Transversal normal-form data for an amalgamated free product

Definition

Let $K$ be embedded in $G$ and $H$ as in Free products with amalgamation along monomorphisms. By The Axiom of Choice, choose left-coset transversals $S_G,S_H$ containing the identity. A normal word is $s_1\cdots s_nk,$ where $n\in\mathbb N$ (The natural numbers $\mathbb{N}$ (von Neumann)), $k\in K$ , every $s_j$ is a nonidentity representative from $S_G$ or $S_H$ , and consecutive representatives come from different factors. Length zero means the word is just $k$ . The written form depends on the transversals.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 results over 16 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

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)