Tietze Transformations

Tietze Transformations

Given a group $G$ there are infinitely many group presentations for $G$. If we have a group presentation for $G$ then we can perform what are called "Tietze transformations" in order to get a new group presentation of the same group. We state the Tietze transformations below:

Let $G = \langle a, b, ... : P, Q, ... \rangle$ be a group presentation of $G$.

• Transformation 1 (Inserting a Relation): If $U$ is a relation that can be obtained from $P$, $Q$, …, then insert $U$:
(1)
\begin{align} \quad G = \langle a, b, ... : P, Q, ..., U \rangle \end{align}
• Transformation 2 (Deleting a Relation): If $U$ is a relation in $\{ P, Q, ... \}$ that is a consequence of $P$, $Q$, …, then delete $U$.
• Transformation 3 (Inserting a Generator): If $x$ is a generator that can be expressed in terms of the other generators $a, b, ...$ by a relation $x = W(a, b, ... )$, then insert $x$:
(2)
\begin{align} \quad G = \langle a, b, ..., x : P, Q, ..., W(a, b, ...) \rangle \end{align}
• Transformation 4 (Deleting a Generator): If $x$ is a generator in $\{ a, b, ... \}$ then delete $x$ and replace all instances of $x$ in the relations $\{ P, Q, ... \}$.
 Definition: Two group presentations are said to be Equivalent if one can be transformed into the other by Tietze transformations.

Let's look at an example of a Tietze transformation.

Example 1

Use Tietze transformations to transform the group presentation $\langle a, b, c : b^3 = 1, (bc)^3 = 1 \rangle$ into the group presentation $\langle x, y, z : x^3 = 1, y^3 = 1 \rangle$.

We start with the first group presentation:

(3)
\begin{align} \quad \langle a, b, c : b^3 = 1, (bc)^3 = 1 \rangle \end{align}

Introduce a new variable $y$ with the relation $y = bc$ (3):

(4)
\begin{align} \quad \langle a, b, c, y : b^3 = 1, (bc)^3 = 1, y = bc \rangle \end{align}
(5)
\begin{align} \quad \langle a, b, c, y : b^3 = 1, (bc)^3 = 1, c = b^{-1} y \rangle \end{align}

Delete the variable $c$ (4):

(6)
\begin{align} \quad \langle a, b, y : b^3 = 1, y^3 = 1, \rangle \end{align}

Introduce a new variable $x$ with the relation $x = b$ (3):

(7)
\begin{align} \quad \langle a, b, x, y : b^3 = 1, y^3 = 1, x = b \rangle \end{align}

Delete the variable $b$ (4):

(8)
\begin{align} \quad \langle a, x, y : x^3 = 1, y^3 = 1, \rangle \end{align}

Introduce a new variable $z$ with the relation $z = a$ (3):

(9)
\begin{align} \quad \langle a, x, y, z : x^3 = 1, y^3 = 1, z = a \rangle \end{align}

Delete the variable $a$ (4):

(10)
\begin{align} \quad \langle x, y, z : x^3 = 1, y^3 = 1 \rangle \end{align}