Tietze transform

Tietze transforms are the following four transformations whereby onecan transform a presentation of a group into another presentation ofthe same group:

1. If a relation $W=V$, where $W$ and $V$ are some word in thegenerators of the group, can be derived from the defining relations ofa group, add $W=V$ to the list of relations.

2. If a relation $W=V$ can be derived from the remaininggenerators, remove $W=V$ fronm the list of relations.

3. If $W$ is a word in the generators and $W=x$, then add $x$ tothe list of generators and $W=x$ to the list of relations.

4. If a relation takes the form $W=x$, where $x$ is a generator and$W$ is a word in generators other than $x$, then remove $W=x$ from thelist of relations, replace all occurences of $x$ in the remainingrelations by $W$ and remove $x$ from the list of generators.

Note that transforms 1 and 2 are inverse to each other and likewise 3and 4 are inverses. More generally, the term “Tietze transform”referes to a transform which can be expressed as the compositon of afinite number of the four transforms listed above. By way ofcontrast, the term “elementary Tietze transformation” is usedto denote the four transformations given above and the term“general Tietze transform” could be used to indicate a memberof the larger class.

Tieze showed that any two presentations of the same finitely presentedgroup differ by a general Tietze transform.