I am new to combinatorial group theory and have a question regarding Tietze transformations. Say I have a group G, finitely presented by $\langle X | R \rangle$, and a sequence of (finitely many) Tietze transformations ending in $\langle X' | R' \rangle$, which in turn finitely presents a group $G'$. Further, say I want to translate a word $w$ (written in $X$ and its inverses) from $\langle X | R \rangle$ to an equivalent word $w'$ in $\langle X' |R' \rangle$ (i. e. a word $w'$ denoting the same group element in $G'$ as $w$ denoted in $G$, where $G \cong G'$).
Since Tietze transformations in general produce isomorphisms, would I just need to know the mapping of the generators of $X$ to the generators of $X'$ in order to aquire an equivalent word (i. e. one of many representations of the equivalent group element in $G'$ written in $X'$ and its inverses)? How would I need to account for a discrepancy in the numbers of generators in both presentations, in order to translate $w$ correctly?
Or to break down my question into single Tietze steps:
- When adding a new generator $y$, say to $X = \{ x_1, x_2, ... , x_n \}$ what do I map to the new generator $y$? As I understand, $y$ can be written in the generators $x_1, x_2, ..., x_n$ (or their inverses). Would it be necessary to map the preimage of y to y in order to translate the word $w$? I would assume it only being necessary, if the letter $y$ itself is to be part of the translation, since $y$ itself can be written in $x_1, x_2, ..., x_n$ (or their inverses)?
- When cancelling a generator, say $y$ (assuming there exists the relevant relator, say $ys^-1$, in $R$ to do so), I would need to map $y$ to $s$ ($s$ being written in $x_1, x_2, ...$ or their inverses). So I would need a map from all $x_i$ to $x_i$ as well as from $y$ to $s$.
- When adjusting only $R$ with a Tietze transformation the normal closure stays the same. So I would assume there is no mapping necessary for a translation of $w$ .
My question being, is my understanding correct, or am I missing something (or several things)?