_{2}respectively and I know that B is torsion-free. Is this enough information to find B? From what I can find on wikipedia, it seems that, in general, you need the SES to split in order to have a nice presentation of B in terms of A and C, however this would give B being equal to Z+Z

_{2}which has torsion, so I guess my sequence doesn't split. I have some heuristic evidence which suggests that B is isomorphic to Z, which obviously fits in to the SES. For those interested, I'm trying to show that the braid group on two strings (B) is isomorphic to the integers, given that I've calculated the pure 2-braid group (A). If this isn't enough information, I guess I'll have to appeal to the homomorphisms in the sequence and chase some elements.