Prove that the group $G$ with generators $x,y,z$ and relations $z^y=z^2$, $x^z=x^2$, $y^x=y^2$ has order $1$.
This is a problem on Page 56 of Derek J.S. Robinson's A Course in the Theory of Groups (GTM 80). I think $z^y$ means the result of $y$ acting on $z$, and may be defined as $y^{-1}zy$.
Suppose that $F$ is a free group generated by $x,y,z$. The epimorphism $\pi: F \rightarrow G$ has its kernel $K$ generated by $z^yz^{-2}$, $x^zx^{-2}$, $y^xy^{-2}$. How to prove $K=F$? Or, how to prove $x,y,z \in K$? I've tried but didn't find the right way.
Thank you very much.