Who says there is no homom.? There is always the trivial homom.: send everything to the unity element. But for this there is no more since:
If there was a non-trivial hom. F: S_3 --> Z_3 it would have to be onto ==> Then S_3/Ker F ~ Z_3, but this'd mean S_3 has a normal sbgp. (namely, ker F) of order 2, which isn't true.