  In this note we derive an upper bound on the number of 2-spheres in the fixed
point set of a smooth and homologically trivial cyclic group action of prime
order on a simply-connected 4-manifold. This improves the a priori bound which
is given by one half of the Euler characteristic of the 4-manifold. The result
also shows that in some cases the 4-manifold does not admit such actions of a
certain order at all or that any such action has to be pseudofree.
