# Tietze transform

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

1. 1.

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

2. 2.

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

3. 3.

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

4. 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 the list of relations, replace all occurences of $x$ in the remaining relations by $W$ and remove $x$ from the list of generators.

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

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

Title Tietze transform TietzeTransform 2013-03-22 15:42:40 2013-03-22 15:42:40 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Definition msc 20F10 elementary Tietze transformation general Tietze transform