HNN extension

Summary

An HNN extension of a group $G$ adds one new element $t$. This element conjugates a subgroup $H$ of $G$ onto another subgroup $K$ by a given isomorphism. The group $G$ embeds in the new group, so two isomorphic subgroups of $G$ become conjugate in a larger group. The construction is named after Graham Higman, Bernhard Neumann and Hanna Neumann. Together with the free product with amalgamation, it is a basic way to build groups from smaller ones. It is used in the proof of the Higman embedding theorem.