Let M be a closed four-manifold whose first homology group is nontrivial and free abelian. If the Euler characteristic is nonzero and the second Betti number is non-two, then the only groups which can admit homologically trivial, locally linear actions on M are cyclic. Loosely speaking, this result (and some of my earlier work) helps make precise the intuitively plausible notion that "most" four-manifolds have small symmetry groups. The main tool is Borel equivariant cohomology, a functor which combines information about a space and a group acting on it using group cohomology. I will provide background and context that I hope will be appropriate to topologists who do not specialize in group actions.