We study the pure braid group short exact sequence described by Fadell and Neuwirth, namely $$0 \to P_{m-n}(RP^2-\{x_1,...,x_n\}) \to P_m(RP^2) \to P_n(RP^2) \to 0$$ and the torsion of the pure Braid groups $P_n(RP^2)$ and of the braid groups $B_n(RP^2)$. The short exact sequence for $n=2$ and $m=3$ splits. This was was shown by Burskirk in the 60's. It is an open question the cases where $m>3$. We show that the sequence does not splits if $m>3$. For the torsion we show that there is a torsion element of $P_n(RP^2)$ of order k if and only if k is either 2 or 4. Similar there is a torsion element of $B_n(RP^2)$ of order k if and only if k divides either 4n or 4(n-1). Also the only element of order 2 in $B_n(RP^2)$ is the full twist. As a consequence of our result we can show that a k-th root of the full twist exists if and only if k divides either 2n or 2(n-1). For the non-splitting result we use some approach of coincidence theory. For the study of the torsion we use techniques of fibrations more standard in the study of the braids.