Show that $\langle x,y\mid xyx=yxy,x^3=y^2\rangle\cong\{e\}$.
So we are trying to show that $\langle x,y\mid xyx=yxy,x^3=y^2\rangle$ is isomorphic to the trivial group which contains only the identity.
The only approach we can use by manipulating the relations $xyx=yxy,x^3=y^2$. For example, from $xyx=yxy$ we can get $xyxy=yx$. But I am clueless on how can we relate it to the trivial group?
My questions are:
1. How can we show the isomorphism?
2. I googled this statement and the Andrews-Curtis Conjecture appears quite often, how does it relate to the conjecture? Is it potential counterexample to the conjecture?
Just to note: I have just learned some topics related to group generators and relations, but not the Todd-Coxeter algorithm yet, so I might not be able to use some advanced tricks.
Helps are greatly appreciated. Many thanks!