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I've been trying to google this question, but to no avail. The question sounds elementary but I hope it's suitable for the experts at MO!

Let $B_n$ be the braid group on $n$-strands and $P_n$ be the pure braid group on $n$-strands. We have a well know decomposition of $P_n$ as $F_n \rtimes P_{n-1}$. In this light, my question is simply:

Are there known decompositions of $B_n$ in terms of direct/semidirect products of $B_{k}$ for $1\leq k < n$ and (possibly) other subgroups of $B_n$

I'd even be interested for special cases of $n$


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$B_3$ is a central extension of $PSL_2(\mathbb Z)$. That's a fairly standard one. –  Ryan Budney Jul 12 '12 at 19:29
Every braid group splits as a semidirect product: simply map each (standard) generator to $1\in\mathbb{Z}$. –  Steve D Jul 12 '12 at 21:29

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