DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

The transversal data used for HNN normal forms

Definition

Work in the associated-subgroup presentation

$G = ⟨ A, t ∣ t c t^{- 1} = ϕ (c) for c \in C_{-} ⟩$

from The edge-group presentation is equivalent to the associated-subgroup presentation. Choose right-coset transversals

$S_{-} \subseteq A for C_{-} \ A, S_{+} \subseteq A for C_{+} \ A,$

each containing the identity; equivalently, every $a \in A$ has unique decompositions

$a = c s (c \in C_{-}, s \in S_{-}), a = d r (d \in C_{+}, r \in S_{+}) .$

An HNN word

$a_{0} t^{ε_{1}} a_{1} \dots t^{ε_{n}} a_{n}$

is in transversal normal form relative to $(S_{-}, S_{+})$ when

$a_{i} \in S_{-} if ε_{i} = 1, a_{i} \in S_{+} if ε_{i} = - 1,$

for each $1 \leq i \leq n$ , and whenever $ε_{i} = - ε_{i + 1}$ the coefficient $a_{i}$ is not the identity. The initial coefficient $a_{0}$ is arbitrary in $A$ .

Depends on

Used by

FALSE: HNN normal form is canonical without choosing transversals False statement
Normal forms in an HNN extension are unique relative to chosen transversals 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

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory (standard reference, not scraped)