On finite groups acting on homology 4-spheres and finite subgroups of SO(5)

We show that a finite group which admits a faithful, smooth, orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by explicitly determining the various groups which can occur (up to an indeterminacy of index two in the case of solvable groups). As a consequence we obtain also a characterization of the finite groups which are isomorphic to subgroups of the orthogonal groups SO(5) and O(5).