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In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the `full twist' braid. Our main results may be summarised as follows: first, the pure braid group short exact sequence\n  1 --> P_{m-n}(RP^2 - {x_1,...,x_n}) --> P_m(RP^2) --> P_n(RP^2) --> 1\n does not split if m > 3 and n=2,3. Now let n > 1. 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