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

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The associated homomorphism from an HNN extension to the integers

Definition

Let

$G = ⟨ A, t | t α (c) t^{- 1} = β (c) for c \in C ⟩$

be an HNN extension. The defining relators send every element of $A$ to $0$ and the stable letter $t$ to $1 \in Z$ , so Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group yields a unique group homomorphism

$χ_{G} : G \to Z$

with

$χ_{G} (a) = 0 (a \in A), χ_{G} (t) = 1.$

This is the associated homomorphism to the integers, or the exponent-sum map of the stable letter.

Depends on

Used by

Nothing in the library uses this result yet.

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

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