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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 $\mathbb{Z} \times \mathbb{Z}$.

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    Are any powers of $y$ in $H=\langle x, (xy)^3\rangle$? That would imply that the index is infinite. I don't remember how that nifty representation of $\mathcal{B}_3$ goes again, so can't prove it now.2011-07-01

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