Tietze Transformations Example 2

# Tietze Transformations Example 2

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 2

Use Tietze transformations to transform the group presentation $\langle a, b : a^2 = 1, (ab)^2 = 1 \rangle$ into the group presentation $\langle x, y : x^2 = 1, y^2 = 1 \rangle$.

(1)
\begin{align} \quad \langle a, b : a^2 = 1, (ab)^2 = 1 \rangle \end{align}

Introduce a new variable $y$ with the relation that $y = ab$:

(2)
\begin{align} \quad \langle a, b, y: a^2 = 1, (ab)^2 = 1, y = ab \rangle \end{align}
(3)
\begin{align} \quad \langle a, b, y : a^2 = 1, y^2 = 1, b = a^{-1}y \rangle \end{align}

Delete the variable $b$:

(4)
\begin{align} \quad \langle a, y : a^2 = 1, y^2 = 1 \rangle \end{align}

Introduce a new variable $x$ with the relation that $x = a$:

(5)
\begin{align} \quad \langle a, x, y : a^2 = 1, y^2 = 1, x = a \rangle \end{align}
(6)
\begin{align} \quad \langle a, x, y : x^2 = 1, y^2 = 1, x = a \rangle \end{align}

Delete the variable $a$:

(7)
\begin{align} \quad \langle x, y : x^2 = 1, y^2 = 1\rangle \end{align}