Group generated by elements with a relation

Question

Could someone show me how to prove that the group generated by $x,y,z$ with the single relation $yxyz^{-2}=1$ is actually a free group.

Answer 1

Let $G$ be your group and $F$ the free group with two generators $u,v$. You have a homomorphism $\phi\colon F\to G$ given by $u\mapsto y, v\mapsto z$. And you have a homomorphism $\psi\colon G\to F$ given by $x\mapsto???$, $y\mapsto u$, $z\mapsto v$ (you just need to check that $\psi(y)\psi(x)\psi(y)\psi(z)^{-2}=1$). Show that $\psi\circ \phi=\operatorname{id}_F$ and $\phi\circ \psi=\operatorname{id}_G$.