In group theory, Tietze transformations are used to transform a given presentation of a group into another, often simpler presentation of the same group. These transformations are named after Heinrich Tietze who introduced them in a paper in 1908.[1]
A presentation is in terms of generators and relations; formally speaking the presentation is a pair of a set of named generators, and a set of words in the free group on the generators that are taken to be the relations. Tietze transformations are built up of elementary steps, each of which individually rather evidently takes the presentation to a presentation of an isomorphic group. These elementary steps may operate on generators or relations, and are of four kinds.