HNN extension

In mathematics, the HNN extension is an important construction of combinatorial group theory.

Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group G into another group G' , in such a way that two given isomorphic subgroups of G are conjugate (through a given isomorphism) in G' .

Construction

Let G be a group with presentation

G

=

⟨

S

∣

R

⟩

{\displaystyle G=\langle S\mid R\rangle }

, and let

α

:

H

→

K

{\displaystyle \alpha \colon H\to K}

be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define

G

∗

α

=

⟨

S

,

t

∣

R

,

t

h

t

−

1

=

α

(

h

)

,

∀

h

∈

H

⟩

.

{\displaystyle G*_{\alpha }=\left\langle S,t\mid R,tht^{-1}=\alpha (h),\forall h\in H\right\rangle .}

The group

G

∗

α

{\displaystyle G*_{\alpha }}

is called the HNN extension of G relative to α. The original group G is called the base group for the construction, while the subgroups H and K are the associated subgroups. The new generator t is called the stable letter.

Key properties

Since the presentation for

G

∗

α

{\displaystyle G*_{\alpha }}

contains all the generators and relations from the presentation for G, there is a natural homomorphism, induced by the identification of generators, which takes G to

G

∗

α

{\displaystyle G*_{\alpha }}

. Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into

G

∗

α

{\displaystyle G*_{\alpha }}

. A consequence is that two isomorphic subgroups of a given group are always conjugate in some overgroup; the desire to show this was the original motivation for the construction.

Britton's Lemma

A key property of HNN-extensions is a normal form theorem known as Britton's Lemma.