# Finding a normal form for the group given by $\langle a,t| t^{-1}at = a^{2}\rangle.$

Let $$G$$ be a group with presentation $$\langle a,t| t^{-1}at = a^{2}\rangle$$

I want to find a normal form for $$G$$. That is, a collection of words on the alphabet $$\{a,t\}$$ such that they represent the elements of $$G$$ in a unique way. Things that I know:

·) Every element of $$G$$ can be represented as $$t^{n}a^{k}t^{-m}$$ for $$n,m,k \in \mathbb{Z}$$ with $$n, m \geq 0$$.

·) There's a well-defined monomorphism

$$\begin{array}{rcl} G & \hookrightarrow & GL_{2}(\mathbb{Q})\\ a & \longmapsto & \begin{pmatrix} 1 & 1\\ 0 & 1\\ \end{pmatrix} =: A\\ t & \longmapsto & \begin{pmatrix} \dfrac{1}{2} & 0\\ 0 & 1\\ \end{pmatrix} =: T\\ \end{array}$$ Then, $$t^{n}a^{k}t^{-m} = t^{n'}a^{k'}t^{-m'}$$ iff $$T^{n}A^{k}T^{-m} = T^{n'}A^{k'}T^{-m'}$$, which is equivalent to say $$m - n = m^{'} - n^{'} \text{ and } \dfrac{k}{2^{n}} = \dfrac{k^{'}}{2^{n^{'}}}$$ From this, how can I get a list of diferent words and conclude that every word in $$G$$ is one of these words, i.e. remove the redundant words and find the normal form?

• Useful: Britton's Lemma. Commented Oct 24, 2019 at 17:34

You need to add a criterion to insure uniqueness. For example, consider the "choice" function $$\phi$$ such that for any word $$w \in G$$ : $$\phi(w) = \min\{n \geq 0 ~|~ \exists m\geq 0, k\in \mathbb{Z} : t^n a^k t^{-m} = w\}$$
This function is well defined from your first bullet point. Any word has a unique reduced word attached to it : if $$w = t^n a^k t^{-m}$$ then we also have $$w = t^{\phi(w)}a^{k2^{\phi(w)-n}}t^{n-m-\phi(w)}$$ which is a form that does not depend on $$n$$, $$m$$ or $$k$$ from your second bullet point.