Why are these two groups isomorphic?

Question

$\langle x, y \mid x^7 = e, y^3 = e, yxy^{-1} = x^2 \rangle$ and $\langle x, y \mid x^7 = e, y^3 = e, yxy^{-1} = x^4 \rangle$. Is it because $x^4 = (x^2)^2$, or is this the wrong reason why they are the same?

Answer 1

Set $G=\langle x,y\mid x^7=e,y^3=e,yxy^{-1}=x^2\rangle$.

Now, in $G$, we have $$y^2xy^{-2}=y(yxy^{-1})y^{-1}=yx^2y^{-1}=yxy^{-1}yxy^{-1}=x^2x^2=x^4.$$ Since $G$ is generated by $x$ and $y^{-1}$, writing $z=y^{-1}$ gives a presentation $$G=\langle x,z\mid x^7=e,z^3=e,zxz^{-1}=x^4\rangle.$$ This shows the groups in your post are the same.