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Group theory and generalizations Definition

HNN extension

Builds on GroupSubgroupGroup isomorphismPresentation of a groupFree productNormal subgroup

Definition

Let GG be a group with a presentation ⟨S∣R⟩\langle S \mid R \rangle. Let HH and KK be subgroups of GG, and let α ⁣:H→K\alpha \colon H \to K be a group isomorphism. Let tt be a new symbol not in SS. The HNN extension of GG relative to α\alpha is the group G∗α=⟨S,t∣R, tht−1=α(h) for all h∈H⟩,G *_{\alpha} = \langle S, t \mid R,\ t h t^{-1} = \alpha(h) \text{ for all } h \in H \rangle, where each h∈Hh \in H and α(h)∈K\alpha(h) \in K is written as a word in SS. Equivalently, G∗αG *_{\alpha} is the quotient of the free product G∗⟨t⟩G * \langle t \rangle, with ⟨t⟩\langle t \rangle infinite cyclic, by the normal subgroup generated by the elements tht−1α(h)−1t h t^{-1} \alpha(h)^{-1} for h∈Hh \in H. The element tt is the stable letter, and HH and KK are the associated subgroups.

Remarks

The natural map from GG to G∗αG *_{\alpha} is injective, so GG is a subgroup of G∗αG *_{\alpha}; this is a theorem of Higman, Neumann and Neumann. Britton's lemma describes when a word in SS and tt represents the identity.

Example: let G=⟨a∣ ⟩≅ℤG = \langle a \mid \ \rangle \cong \Z, H=⟨am⟩H = \langle a^m \rangle and K=⟨an⟩K = \langle a^n \rangle with m,nm, n nonzero integers, and α(am)=an\alpha(a^m) = a^n. The map α\alpha is an isomorphism, since both subgroups are infinite cyclic and α\alpha sends a generator to a generator. The HNN extension is ⟨a,t∣tamt−1=an⟩\langle a, t \mid t a^m t^{-1} = a^n \rangle, the Baumslag–Solitar group BS(m,n)BS(m, n). For m=n=1m = n = 1 it is ⟨a,t∣tat−1=a⟩≅ℤ2\langle a, t \mid t a t^{-1} = a \rangle \cong \Z^2.

Non-example: the rule tht−1=φ(h)t h t^{-1} = \varphi(h) for a homomorphism φ ⁣:H→K\varphi \colon H \to K that is not injective is not an HNN extension. If φ(h)=1\varphi(h) = 1 for some h≠1h \ne 1, then h=t−1φ(h)t=1h = t^{-1} \varphi(h) t = 1 in the quotient, so GG does not embed.

Reading path

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Free product, concept, no evidence levelFree productconceptGroup, concept, no evidence levelGroupconceptGroup isomorphism, concept, no evidence levelGroup isomorphismconceptNormal subgroup, concept, no evidence levelNormal subgroupconceptPresentation of a group, concept, no evidence levelPresentation of a gro…conceptSubgroup, concept, no evidence levelSubgroupconceptHNN extension, definition, no evidence levelHNN extensiondefinition
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  1. Free product concept
  2. Group concept
  3. Group isomorphism concept
  4. Normal subgroup concept
  5. Presentation of a group concept
  6. Subgroup concept
  7. HNN extension definition

Further reading

  1. Web https://en.wikipedia.org/wiki/HNN_extension
About this page
TagTACX13XZST
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Kinddefinition
WritingW0 stub
MSC20E06, 20F05
DefinesHNN extension
UsesGroup, Subgroup, Group isomorphism, Presentation of a group, Free product, Normal subgroup
Changed2 November 2026

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