  We show that a closed orientable 3-manifold M admits an action of Z_p with
fixed point set S^1 iff M can be obtained as the result of surgery on a
p-periodic framed link L and Z_p acts freely on the components of L. We prove a
similar theorem for free Z_p-actions. As an interesting application, we prove
the following, rather unexpected result: for any M as above and for any odd
prime p, H_1(M, Z_p)\ne Z_p. We also prove a similar criterion of 2-periodicity
for rational homology 3-spheres.
