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

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. 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.

## Mathematics Subject Classification

20F10*no label found*

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