  Generalising previous results on classical braid groups by Artin and Lin, we
determine the values of m, n $\in$ N for which there exists a surjection
between the n-and m-string braid groups of an orientable surface without
boundary. This result is essentially based on specific properties of their
lower central series, and the proof is completely combinatorial. We provide
similar but partial results in the case of orientable surfaces with boundary
components and of non-orientable surfaces without boundary. We give also
several results about the classification of different representations of
surface braid groups in symmetric groups.
