A short tutorial on Tietze transformations

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:

Type I Transformation: The presentation
$$\langle~ S~|~W~\rangle$$ is equivalent to the presentation
$$\langle~ S, x~|~W\cup \{x^{-1}w\}~\rangle$$ where $x\not\in S$ and $w$ is any word whose symbols belong to $S$. Basically, we are just introducing shorthand for a string of symbols already in $S$. It is like saying, “let $x$ equal this string of symbols”. It’s really jus syntactic. So, we can modifying presentations by introducing new variables that equal existing string, and if we see a presentation like that we can just remove the new variable and relation.

Type II Transformation: The second kind is adding or removing a relation. Let’s say we have
$$\langle~ S~|~W~\rangle,$$ and $w$ is in the normal closure of $W$. Then we can just add it:
$$\langle~ S~|~W\cup \{w\}~\rangle.$$ Similarly, we can remove superfluous relations. We can do both at once by substitution. For example, we can change
$$\langle~ x,y,z~|~ xy^2, z^{-1}xy~\rangle$$ to
$$\langle~ x,y,z~|~zy, z^{-1}xy~\rangle.$$


I used this example earlier:
$$\langle~ x,y~|~xyx=yxy~\rangle\\
\langle~ a,b~|~a^3 = b^2~\rangle$$ Let’s see that they give isomorphic groups. That is, one can be transformed into another via Tietze transformations:

  1. We start out with $\langle~ x,y~|~xyx=yxy~\rangle$
  2. (I) Let us introduce a new variable that equals $xy$:
  3. $$\langle~ x,y,a~|~xyx=yxy, a^{-1}xy~\rangle$$

  4. (II) Now replace all the occurences of $xy$ in the first relation:
  5. $$\langle~ x,y,a~|~ax=ya,a^{-1}xy~\rangle$$

  6. (I) Now we introduce a new variable $b$ equal to $ax$:
  7. $$\langle~ x,y,a,b~|~ax=ya,a^{-1}xy, b^{-1}ax~\rangle$$

  8. (II) Replace $ax$ in the first relation:
  9. $$\langle~ x,y,a,b~|~b=ya,a^{-1}xy, b^{-1}ax~\rangle$$

  10. (II) We know that $y=x^{-1}a$ from the second relation so put that in the first:
  11. $$\langle~ x,y,a,b~|~b=x^{-1}a^2,a^{-1}xy, b^{-1}ax~\rangle$$

  12. (I) Now we see that $y$ is as in the first kind of Tietze transformation. That is, it is just equal to some other stuff, so we can remove it and its relation:
  13. $$\langle~ x,a,b~|~b=x^{-1}a^2, b^{-1}ax~\rangle$$

  14. (II) By the second relation we can replace $x^{-1}$ with $b^{-1}a$ in the first relation:
  15. $$\langle~ x,a,b~|~b=b^{-1}a^3, b^{-1}ax~\rangle$$

  16. (I) Now $x$ is superfluous because it is just equal to some other stuff:
  17. $$\langle~ a,b~|~b=b^{-1}a^3 \rangle$$

  18. (II) Finally, multiply by $b$ on both sides of the remaining relation:
  19. $$\langle~ a,b~|~b^2=a^3 \rangle$$

If you are given two presentations with finitely many generators and relations, and they are isomorphic, then one can be transformed into another this way. However, there is no algorithmic way in general to decide whether this can be done. But for simple examples like this, you should be able to do them by hand.

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