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In 1949, G. Higman, B.H. Neumann and H. Neumann [[#References|[a4]]] proved several famous embedding theorems for groups using a construction later called the HNN-extension. The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also [[Amalgam of groups|Amalgam of groups]]).

The easiest way to define an HNN-group is in terms of presentations of groups.

==Presentation of groups.==
A [[presentation]] of a [[group]] $G$ is a pair $\langle X | R \rangle$ where $R$ is a subset of $F(X)$, the [[free group]] on the set $X$, and $G$ is isomorphic (cf. also [[Isomorphism]]) to the [[quotient group]] $F(X)/N(R)$, where $N(R)$ is the intersection of all [[normal subgroup]]s of $F(X)$ containing $R$. The subgroup $N(R)$ is called the normal closure of $R$ in $F(X)$.

Given an arbitrary group $G$, there is an obvious [[homomorphism]] $\tau_G : F(G) \rightarrow G$ such that $\tau_G(g) = g$ for all $g \in G$. Clearly, $\langle G | \ker \tau_G \rangle$ is a presentation for $G$.

==HNN-extensions.==
Suppose $\mu : A \rightarrow B$ is an isomorphism of subgroups of a group $G$ and $t$ is not in $G$. The HNN-extension of $G$ with respect to $\mu$ has presentation

\begin{equation*} \left\langle G \bigcup \{ t \} : ( \operatorname { ker } ( \tau _ { G } ) ) \bigcup \left\{ t ^ { - 1 } a ^ { - 1 } t \mu ( a ) : \forall a \in A \right\} \right\rangle. \end{equation*}

The generator $t$ is called the stable letter, $G$ the base group and $A$ and $B$ the associated subgroups of this HNN-extension. When $A = G$, the HNN-extension is called ascending.

Shorthand notation for the above group is $\langle G , t : t ^ { - 1 } A t = B , \mu \rangle$ or $G ^ { * } \mu$.

In [[#References|[a4]]] it was shown that the mapping $G \rightarrow G ^ { * } \mu$ taking $g \rightarrow g$ for all $g \in G$ is a [[Monomorphism|monomorphism]]. The rest of the normal form theorem for HNN-extensions was proved by J.L. Britton in 1963 [[#References|[a1]]] (Britton's lemma): Let $g _ { 0 } , \ldots , g _ { n }$ be a sequence of elements of $G$ and let the letter $\epsilon$, with or without subscripts, denote $\pm 1$. A sequence $g_0, t^{\epsilon_1}, g_1, \cdots, t^{\epsilon_n}, g_n$ will be called reduced if there is no consecutive subsequence $t^{-1} , g_{i} , t$ with $g _ {i} \in A$ or $t, g_{i} , t^{-1}$ with $g_i \in B$. For a reduced sequence and $n \geq 1$, the element

$$g_0 t^{\epsilon_1} g_1 \cdots t^{\epsilon_n} g_n$$

of $G _ { \mu } ^ { * }$ is different from the unit element.

In the original reference [[#References|[a4]]], the following theorem is proved: Every group $G$ can be embedded in a group $G ^ { * }$ in which all elements of the same order are conjugate (cf. also [[Conjugate elements|Conjugate elements]]). In particular, every torsion-free group can be embedded in a group $G ^ { * * }$ with only two conjugacy classes. If $G$ is countable, so is $G ^ { * * }$. Also, every countable group $C$ can be embedded in a group $G$ generated by two elements of infinite order. The group $G$ has an element of finite order $n$ if and only if $C$ does. If $C$ is finitely presentable, then so is $G$.

For an excellent account of the history of HNN-extensions, see [[#References|[a2]]]. See [[#References|[a5]]], Chap. IV, for basic results and landmark uses of HNN-extensions, such as: the torsion theorem for HNN-extensions; the Collins conjugacy theorem for HNN-extensions; the construction of finitely-presented non-Hopfian groups (in particular, the [[Baumslag–Solitar group|Baumslag–Solitar group]] $\langle b , t : t ^ { - 1 } b ^ { 2 } t = b ^ { 3 } \rangle$ is non-Hopfian; cf. also [[Non-Hopf group|Non-Hopf group]]); decompositions of $1$-relator groups; Stallings' classification of finitely-generated groups with more than one end in terms of amalgamated products and HNN-extensions; and Stallings' characterization of bipolar structures on groups.

HNN-extensions are of central importance in, e.g., the modern version of the Van Kampen theorem (based on topological results in [[#References|[a8]]], [[#References|[a9]]]); the Bass–Serre theory of groups acting on trees and the theory of graphs of groups (see [[#References|[a7]]]); Dunwoody's accessibility theorem [[#References|[a3]]]; and JSJ decompositions of groups [[#References|[a6]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  J.L. Britton,   "The word problem"  ''Ann. of Math.'' , '''77'''  (1963)  pp. 16–32</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  B. Chandler,   W. Magnus,   "The history of combinatorial group theory: A case study in the history of ideas" , ''Studies History Math. and Phys. Sci.'' , '''9''' , Springer  (1982)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M.J. Dunwoody,   "The accessibility of finitely presented groups"  ''Invent. Math.'' , '''81'''  (1985)  pp. 449–457</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G. Higman,   B.H. Neumann,   H. Neumann,   "Embedding theorems for groups"  ''J. London Math. Soc.'' , '''24'''  (1949)  pp. 247–254; II.4, 13</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R. Lyndon,   P. Schupp,   "Combinatorial group theory" , Springer  (1977)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E. Rips,   Z. Sela,   "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition"  ''Ann. of Math. (2)'' , '''146''' :  1  (1997)  pp. 53–109</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J.P. Serre,   "Arbres, amalgams, $S L _ { 2 }$"  ''Astéerisque'' , '''46'''  (1977)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E.R. Van Kampen,   "On the connection between the fundamental groups of some related spaces"  ''Amer. J. Math.'' , '''55'''  (1933)  pp. 261–267</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  E.R. Van Kampen,   "On some lemmas in the theory of groups"  ''Amer. J. Math.'' , '''55'''  (1933)  pp. 268–273</td></tr></table>