Consider the braid group $\mathcal{B}_3$ on three strands. It is known that $\mathcal{B}_3 = \langle x,y | xyx = yxy \rangle$ and that the center $Z\mathcal{B}_3$ is infinite cyclic generated by $(xy)^3$.
(1) What is the group $\mathcal{B}_3$ modulo the normal closure of $\langle x,(xy)^3 \rangle$ ?
(2) What is the index of $\langle x,(xy)^3 \rangle$ in $\mathcal{B}_3$ ?
It is not hard to show that $\langle x,(xy)^3 \rangle$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$.