Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28
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An HNN extension realises two chosen isomorphic subgroups as conjugate

Example

If α,β:C→A are injective, then in the HNN extension

G=⟨A,t | t α(c) t−1=β(c) for c∈C⟩,

the subgroups α(C) and β(C) become conjugate by the stable letter:

t α(C) t−1=β(C).

Facts & Assumptions

Given: The HNN extension in the statement.

[L1]

The defining relations of an HNN extension identify t α(c) t−1 with β(c) for every c∈C. (An HNN extension with its stable letter)

[L2]

The stable letter is the universal element that enforces the required conjugacy relation. (The universal property of an HNN extension)

Verification

technique · direct
1.1

For every c∈C, [L1] gives t α(c) t−1=β(c). Hence t α(C) t−1⊆β(C).

L1given
2.1

Applying the same relation to t−1 shows α(c)=t−1β(c)t for every c∈C, so β(C)⊆t α(C) t−1. Thus the two subgroups are conjugate exactly as [L2] predicts.

L1L2step 1.1algebra∎

Depends on

Used by

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