Tietze transformations

These transformations are named after Heinrich Tietze who introduced them in a paper in 1908.

Let G=〈 x | x3=1 〉 be a finite presentation for the cyclic group of order 3.

In order to do this it is necessary to replace all occurrences of the removed generator with its equivalent word.

Let G = 〈 x,y | x3 = 1, y2 = 1, (xy)2 = 1 〉 be a presentation for the symmetric group of degree three.

Through Tietze transformations this presentation can be converted to G = 〈 y, z | (zy)3 = 1, y2 = 1, z2 = 1 〉, where z corresponds to (1,2).