I am working on this one:
Let $G=\langle x,y|xyx^{-1}y^2=1, yx^{-1}yx^2y=1\rangle$. Show that $G\cong\mathbb Z_3$
What I have done for this is to consider subgroup $H=\langle x\rangle$ and to find the index of $H$ in $G$ by doing Todd-Coxeter Algorithm. I did this algorithm and find $3$ right cosets for the subgroup $H$ in the group. That means to me that $[G:H]=3$. I am not sure this made the group to be cyclic, because I know some non abelian semi-direct product. Thanks for helping me in this problem.