HNN extension

In mathematics, the HNN extension is an important construction of combinatorial group theory.Britton's Lemma. If w = 1 in G∗α thenBritton's Lemma (alternate form). If w is such that 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' . Let G be a group with presentation G = ⟨ S ∣ R ⟩ {displaystyle G=langle Smid R angle } , and let α : H → K {displaystyle alpha colon H o K} be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define 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. 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. A key property of HNN-extensions is a normal form theorem known as Britton's Lemma. Let G ∗ α {displaystyle G*_{alpha }} be as above and let w be the following product in G ∗ α {displaystyle G*_{alpha }} :