Introduced in a 1949 paper Embedding Theorems for Groups[1] 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, and let be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define
The group is called the HNN extension ofGrelative 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 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 . Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into . 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.[2] Let be as above and let w be the following product in :
Then Britton's Lemma can be stated as follows:
Britton's Lemma. If w = 1 in G∗α then
either and g0 = 1 in G
or and for some i ∈ {1, ..., n−1} one of the following holds:
εi = 1, εi+1 = −1, gi ∈ H,
εi = −1, εi+1 = 1, gi ∈ K.
In contrapositive terms, Britton's Lemma takes the following form:
Britton's Lemma (alternate form). If w is such that
either and g0 ≠ 1 ∈ G,
or and the product w does not contain substrings of the form tht−1, where h ∈ H and of the form t−1kt where k ∈ K,
then in .
Consequences of Britton's Lemma
Most basic properties of HNN-extensions follow from Britton's Lemma. These consequences include the following facts:
The natural homomorphism from G to is injective, so that we can think of as containing G as a subgroup.
Every element of finite order in is conjugate to an element of G.
Every finite subgroup of is conjugate to a finite subgroup of G.
If contains an element such that is contained in neither nor for any integer , then contains a subgroup isomorphic to a free group of rank two.
^Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN978-3-540-41158-1; Ch. IV. Free Products and HNN Extensions.
^Warren Dicks; M. J. Dunwoody. Groups acting on graphs. p. 14. The fundamental group of graphs of groups can be obtained by successively performing one free product with amalgamation for each edge in the maximal subtree and then one HNN extension for each edge not in the maximal subtree.