Here is a little tutorial on how to use Tietze transformations. They were named after Austrian mathematician Heinrich Franz Friedrich Tietze.

A presentation is a set of generators and relations given by the notation

$$\langle~ S~|~W~\rangle$$ where $S$ is set and $W$ is a set of words in the symbols of $S$. $S$ is called the set of generators and $W$ is called the set of relations or relators. For example,

$$\langle~ r,s~|~rsr^{-1}s^{-1}~\rangle.$$ The group $G$ given by such a presentation is the quotient of the free group $F$ on $S$ by the normal closure of the set $W$ in $F$. In our example, $G\cong \Z\times\Z$. Sometimes we may abuse notation and write something like $xy = x^2$ instead of $xyx^{-2}$ in the set $W$.

We may be given *two* presentations such as:

$$\langle~ x,y~|~xyx=yxy~\rangle\\

\langle~ a,b~|~a^3 = b^2~\rangle$$ and we need to know whether the corresponding groups are isomorphic. To do this, we can perform a set of moves or steps, transforming each presentation into a new presentation whose group is isomorphic to the group of the previous.

There are two basic types of transformation: adding or removing generators, and adding or removing relations:

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