I have a question in which I have to transform
$\textbf{I.}$ $\langle a,b,c \mid b^2, (bc)^2\rangle$
to
$\textbf{II.}$ $\langle x,y,z\mid y^2, z^2\rangle$
using Tietze transformations.
My attempt- $\langle a,b,c \mid b^2, (bc)^2\rangle \overset{1}{\rightarrow}\langle b,c \mid b^2, (bc)^2\rangle \overset{2}{\rightarrow}\langle x,b,c \mid b^2, (bc)^2\rangle \overset{3}{\rightarrow}\langle x,y,b,c \mid y^{-1}b, b^2, (bc)^2\rangle \overset{4}{\rightarrow}\langle x, y, c \mid y^2, (yc)^2\rangle \overset{5}{\rightarrow}\text{ ?} $
Is my step $4$ correct, can I do it as I have defined $y$ as $b$ so I can remove $b$ and replace it by $y$? If it is correct what should be my step $5$?
$$\begin{align}\langle a,b,c|\ b^2,(bc)^2\rangle&=\langle a,b,c,z|\ b^2,z^2, b^{-1}z=c\rangle\\&=\langle a,b,z|\ b^2,z^2\rangle\end{align}$$
In the first step we added a generator $z=bc$. Then the relation $(bc)^2=1$, becomes $z^2=1$, and need to put the relation $z=bc$, which we write as $b^{-1}z=c$.
In the second step we remove the generator $c$, taking away the relation $c=b^{-1}z$, which only defines $c$.