PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The universal property of an HNN extension

Statement

Let

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

be an HNN extension. Let $H$ be a group, let $f : A \to H$ be a group homomorphism, and let $h \in H$ satisfy

$h f (α (c)) h^{- 1} = f (β (c)) (c \in C) .$

Then there is a unique group homomorphism

$\overline{f} : G \to H$

whose restriction to $A$ is $f$ and whose value on the stable letter is $\overline{f} (t) = h$ .

Facts & Assumptions

Given: The HNN extension, the homomorphism $f : A \to H$ , and the element $h \in H$ in the statement.

[L1]

An HNN extension is the group presented by adjoining a stable letter $t$ and the relators $t α (c) t^{- 1} = β (c)$ for every $c \in C$ . (An HNN extension with its stable letter)

[L2]

A map on generators of a presentation extends uniquely once every defining relator evaluates to the identity. (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group)

Proof

technique · constructive

1.1L1L2givenconstruct

Use $f$ on the base-group generators of a presentation of $A$ and send the stable letter $t$ to $h$ . The old relators from $A$ are satisfied because $f$ is a homomorphism, and each new HNN relator from [L1] is satisfied because the hypothesis gives $h f (α (c)) h^{- 1} = f (β (c))$ .

2.1L2step 1.1discharge-construct∎

Therefore [L2] gives a unique homomorphism $\overline{f} : G \to H$ extending those assignments. By construction it restricts to $f$ on $A$ and sends $t$ to $h$ , which is exactly the required universal property.

Depends on

Used by

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)
C. Loh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)