Tietze Transformations Example 1

Tietze Transformations Example 1

On the Tietze Transformations page we defined Tietze transformations for getting alternative group presentations of a group. We will now look at another example of a Tietze transformation. Another example can be found on the page below:

• Tietze Transformations Example 1

Example 1

Use Tietze transformations to transform the group presentation $\langle x, y : x^3 = 1, y^2 = 1, (xy)^2 = 1 \rangle$ into the group presentation $\langle y, z : (zy)^3 = 1, y^2 = 1, z^2 = 1 \rangle$.

(1)
\begin{align} \langle x, y : x^3 = 1, y^2 = 1, (xy)^2 = 1 \rangle \end{align}

We introduce a new variable $z$ with the relation $z = xy$:

(2)
\begin{align} \quad \langle x, y, z : x^3 = 1, y^2 = 1, (xy)^2 = 1, z = xy \rangle \end{align}
(3)
\begin{align} \quad \langle x, y, z : x^3 = 1, y^2 = 1, z^2 = 1, x = zy^{-1} \rangle \end{align}

We now remove the variable $x$:

(4)
\begin{align} \quad \langle y, z : (zy^{-1})^3 = 1, y^2 = 1, (zy^{-1}y)^2 = 1 \rangle \end{align}
(5)
\begin{align} \quad \langle y, z : (zy^{-1})^3 = 1, y^2 = 1, z^2 = 1 \rangle \end{align}

Lastly, observe that since $y^2 = 1$ we have that $z = zy^{2}$. Therefore $zy^{-1} = zy$, and so:

(6)
\begin{align} \quad \langle y, z : (zy)^3 = 1, y^2 = 1, z^2 = 1 \rangle \end{align}