  A 3-manifold $M$ is said to be $p$-periodic ($p\geq 2$ an integer) if and
only if the finite cyclic group of order $p$ acts on $M$ with a circle as the
set of fixed points. This paper provides a criterion for periodicity of
rational homology three-spheres. Namely, we give a necessary condition for a
rational homology three-sphere to be periodic with a prime period. This
condition is given in terms of the quantum SU(3) invariant. We also discuss
similar results for the Murakami-Ohtsuki-Okada invariant.
