Tietze transform

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

  1. 1.

    If a relationMathworldPlanetmathPlanetmath W=V, where W and V are some word in the generatorsPlanetmathPlanetmathPlanetmath 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 inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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
Canonical name TietzeTransform
Date of creation 2013-03-22 15:42:40
Last modified on 2013-03-22 15:42:40
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Definition
Classification msc 20F10
Defines elementary Tietze transformation
Defines general Tietze transform