Tietze transformations

Summary

Tietze transformations are four elementary moves on a group presentation that do not change the group up to isomorphism. They add or remove a redundant relation, or add or remove a redundant generator. Two finite presentations define isomorphic groups if and only if one can pass from one to the other by finitely many such moves. They are the basic tool to simplify presentations and to recognise when two presentations give the same group.