Tietze transformations are a formalisation of the informal substitution methods that are natural when working with group presentations.
Let be a group presentation, where the ‘specified isomorphism to ’ is unspecified!
The following transformations do not change the group :
T1: Adding a superfluous relation
becomes , where and the normal closure of the relations in the free group on , i.e., is a consequence of ;
T2: Removing a superfluous relation
becomes where , and is a consequence of ;
T3: Adding a superfluous generator
becomes , where , being a new symbol not in , and , where is a word in the other generators, that is is in the image of the inclusion of into ;
T4: Removing a superfluous generator
becomes , where , and with and and no other members of involve .
Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze transformations.
Tietze’s original paper is