I’m reading about Tietze transformation and want to check my understanding. Here’s the statement and an example.
Given a presentation it is possible to add a new generator that is expressed as a word in the original generators. Starting with $G=\left\langle x \mid x^{3}=1\right\rangle$ and letting $y=x^{2}$ the new presentation $G=\left\langle x, y \mid x^{3}=1, y=x^{2}\right\rangle$ defines the same group.
Does this say that, since a word like $x^2y$ from the second presentation can be expressed as $x^2x^2 = x$, it is already included in the group defined by the first presentation? I’m trying to get a setup like that in the First Isomorphism Theorem, i.e. having free groups and homomorphisms $\phi_1: F_{\{x\}} \to G$ and $\phi_2: F_{\{x,y\}} \to G$. What would be ker $\phi_2$ in this case?
Also can we just let $y$ be an arbitrary word? For example, if given a group $G=\left\langle x \mid x^{5}=1\right\rangle$, can we have $y = x^2,x^3,x^4$?